{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Theoretische Informatik - Vorlesung 0 - Teil 2 Mengentheorie\n",
"\n",
"* April 2020\n",
"* Michael Leuschel\n",
"* Lehrstuhl Softwaretechnik und Programmiersprachen\n",
"* Heinrich-Heine Universität Düsseldorf\n",
"\n",
"Grundlagen der Logik und Mengentheorie sind nicht im Skript.\n",
"Hier definieren wir einige Grundlagen und Notationen die im Skript verwendet werden.\n",
"Ein gutes Verständnis dieser Grundlagen und Notationen ist für das Verständnis des Skripts, aber auch anderer Teile der Informatik unumgänglich.\n",
"\n",
"Auswertung der Formeln erfolgt mit dem [Jupyter Kernel](https://gitlab.cs.uni-duesseldorf.de/general/stups/prob2-jupyter-kernel) für [ProB](https://www3.hhu.de/stups/prob)\n",
"\n",
"\n",
"Um dieses Notebook zu starten kann man entweder selber Jupyter und den [ProB Kernel](https://gitlab.cs.uni-duesseldorf.de/general/stups/prob2-jupyter-kernel) installieren.\n",
"Man kann aber auch die Notebooks vom Browser aus mit Binder starten (das dauert besonders beim ersten Mal etwas länger):\n",
"\n",
"[](https://mybinder.org/v2/git/https%3A%2F%2Fgitlab.cs.uni-duesseldorf.de%2Fgeneral%2Fstups%2Fprob-teaching-notebooks/master?filepath=info4%2Fkapitel-0%2FMengentheorie.ipynb)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Mengen\n",
" \n",
"Fundamentale Idee der Mengentheorie: \n",
"\n",
"* _\"Unter einer ‚Menge‘ verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten m unserer Anschauung oder unseres Denkens (welche die ‚Elemente‘ von M genannt werden) zu einem Ganzen.\"_ Georg Cantor (siehe [Textstelle als Bild in Wikipedia](https://de.wikipedia.org/wiki/Datei:Textstelle_mit_der_Mengendefinition_von_Georg_Cantor.png))\n",
"* _\"The ability to regard any collection of objects as a single entity (i.e., as a set).\"_ (Keith Devlin. Joy of Sets. Springer-Verlag.)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"In der Regel gibt es eine Domäne an \"Objekten\" mit denen man Mengen bauen kann.\n",
"Was genau diese Objekte sind interessiert uns in der Mengentheorie nicht.\n",
"\n",
"Fundamental sind diese beiden Symbole:\n",
"* wenn $a$ ein Objekt ist und $x$ eine Menge, dann \n",
" * ist $a \\in x$ wahr, wenn $a$ ein Element von $x$ ist\n",
" * ist $a \\not\\in x$ wahr, wenn $a$ **kein** Element von $x$ ist.\n",
"\n",
"\n",
"$\\in$ und $\\not\\in$ sind Prädikate, verbunden durch die Eigenschaft:\n",
"* $\\forall(a,x).(a\\not\\in x \\Leftrightarrow \\neg(a \\in x))$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Die leere Menge und Gleichheit\n",
"\n",
"Eine besondere Menge ist die leere Menge $\\emptyset$.\n",
"Sie hat keine Elemente:\n",
" * $z = \\emptyset \\Leftrightarrow \\forall(a).(a\\not\\in z)$\n",
"\n",
"Zwei Mengen $x$ und $y$ sind gleich gdw sie die gleichen Elemente haben:\n",
" * $\\forall(x,y).(x=y \\Leftrightarrow \\forall(a).(a\\in x \\Leftrightarrow a \\in y))$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{FALSE}$"
],
"text/plain": [
"FALSE"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"∅ = {1}"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{FALSE}$"
],
"text/plain": [
"FALSE"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{1} = {1,2}"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$"
],
"text/plain": [
"TRUE"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{1,2} = {2,1}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Notationen für Mengen: endliche Enumeration\n",
" \n",
"* explizite Auflistung aller Elemente $\\{a_1,\\ldots,a_n\\}$\n",
"* die Reihenfolge spielt keine Rolle:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$"
],
"text/plain": [
"TRUE"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{2,5,3} = {2,3,5}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Dies ist im Unterschied zu Tupeln und Listen oder Folgen, die oft mit runden und eckigen Klammern geschriben werden:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{FALSE}$"
],
"text/plain": [
"FALSE"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(TRUE,5,3) = (TRUE,3,5) // Vergleich von zwei Tripeln"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{FALSE}$"
],
"text/plain": [
"FALSE"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"[2,5,3] = [2,3,5] // Vergleich von zwei Folgen/Listen"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"* Elemente können in der Enumeration mehrfach auftauchen:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$"
],
"text/plain": [
"TRUE"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{2,5,3,2,5} = {2,3,5}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Vereinigung, Schnitt, Differenz\n",
"\n",
"Drei wichtige Operationen auf Mengen sind wie folgt.\n",
"\n",
"Vereinigung von Mengen $\\cup$ :\n",
" * $z = x\\cup y \\Leftrightarrow \\forall(a).(a\\in z \\Leftrightarrow (a\\in x \\vee a \\in y))$\n",
"\n",
"Schnitt von Mengen $\\cap$:\n",
"* $z = x\\cap y \\Leftrightarrow \\forall(a).(a\\in z \\Leftrightarrow (a\\in x \\wedge a \\in y))$\n",
"\n",
"Differenz von Mengen $\\setminus$:\n",
"* $z = x \\setminus y \\Leftrightarrow \\forall(a).(a\\in z \\Leftrightarrow (a\\in x \\wedge a \\not\\in y))$\n"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{2,3,5,7\\}$"
],
"text/plain": [
"{2,3,5,7}"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{2,3,5} ∪ {5,7}"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{5\\}$"
],
"text/plain": [
"{5}"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{2,3,5}∩{5,7}"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{2,3\\}$"
],
"text/plain": [
"{2,3}"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{2,3,5}−{5,7}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Notationen für Mengen: per Prädikat\n",
" \n",
"Man kann die Elemente einer Menge auch durch ein Prädikat beschreiben:$\\{a \\mid P(a)\\}$.\n",
"Es gilt:\n",
"* $z = \\{a \\mid P(a)\\} \\Leftrightarrow \\forall(a).(a\\in z \\Leftrightarrow P(a))$\n"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{2,3,5\\}$"
],
"text/plain": [
"{2,3,5}"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{a | a>1 & a<6 & a≠4}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Man kann mit dieser Schreibweise auch unendliche Mengen darstellen:"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{\\mathit{a}\\mid \\mathit{a} > 10\\}$"
],
"text/plain": [
"{a∣a > 10}"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{a | a>10}"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{2\\}$"
],
"text/plain": [
"{2}"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{a | a mod 2 = 0} ∩ {2,3,5}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Man kann mit dieser Schreibweise auch die drei Mengenoperationen definieren:\n",
"\n",
"* $x \\cup y = \\{a \\mid a\\in x \\vee a\\in y\\}$\n",
"* $x \\cap y = \\{a \\mid a\\in x \\wedge a\\in y\\}$\n",
"* $x \\setminus y = \\{a \\mid a\\in x \\wedge a\\not\\in y\\}$"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{xy} = \\{2,3,5\\}$\n",
"* $\\mathit{x} = \\{2,3\\}$\n",
"* $\\mathit{y} = \\{2,5\\}$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
"\txy = {2,3,5}\n",
"\tx = {2,3}\n",
"\ty = {2,5}"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = {2,3} & y = {2,5} & xy = {a| a∈x ∨ a∈y}"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{xy} = \\{2\\}$\n",
"* $\\mathit{x} = \\{2,3\\}$\n",
"* $\\mathit{y} = \\{2,5\\}$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
"\txy = {2}\n",
"\tx = {2,3}\n",
"\ty = {2,5}"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = {2,3} & y = {2,5} & xy = {a| a∈x ∧ a∈y}"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{xy} = \\{3\\}$\n",
"* $\\mathit{x} = \\{2,3\\}$\n",
"* $\\mathit{y} = \\{2,5\\}$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
"\txy = {3}\n",
"\tx = {2,3}\n",
"\ty = {2,5}"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = {2,3} & y = {2,5} & xy = {a| a∈x ∧ a∉y}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Als Kürzel führen wir auch die Notation $a..b$ für $\\{x \\mid x \\geq a \\wedge x \\leq b\\}$ ein."
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,2,3,4,5,6,7,8,9,10\\}$"
],
"text/plain": [
"{1,2,3,4,5,6,7,8,9,10}"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1..10"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Ein Theorem (Distributivgesetz 1)\n",
"\n",
"Für alle Mengen $x$, $y$, $z$ gilt:\n",
"* $x \\cup (y \\cap z) = (x\\cup y) \\cap (x\\cup z)$\n",
"\n",
"<img src=\"./img/Venn.svg\" width=\"300\">"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{2,3,44,55\\}$"
],
"text/plain": [
"{2,3,44,55}"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{2,3,55}∪({2,44,77}∩{2,44,66})"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{2,3,44,55\\}$"
],
"text/plain": [
"{2,3,44,55}"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"({2,3,55}∪{2,44,77})∩({2,3,55}∪{2,44,66})"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Beweis von $x \\cup (y \\cap z) = (x\\cup y) \\cap (x\\cup z)$\n",
"\n",
"Für den Beweis benötigen wir diese Lemmata:\n",
"1. $x \\cup y = \\{a \\mid a\\in x \\vee a\\in y\\}$\n",
"2. $x \\cap y = \\{a \\mid a\\in x \\wedge a\\in y\\}$\n",
"3. $a \\in \\{b \\mid P(b)\\} \\equiv P(a)$\n",
"4. $\\phi \\vee (\\psi \\wedge \\rho) \\equiv (\\phi \\vee \\psi) \\wedge (\\phi \\vee \\rho)$\n",
"\n",
"Hier ist ein Äquivalenzbeweis (siehe vorherige Vorlesung): \n",
"1. $m=x \\cup (y \\cap z)$\n",
"2. (Lemma 1) $\\Longleftrightarrow$ $m=\\{a \\mid a\\in x \\vee a\\in (y \\cap z)\\}$ \n",
"3. (Lemma 2) $\\Longleftrightarrow$ $m=\\{a \\mid a\\in x \\vee a\\in \\{b \\mid b\\in y \\wedge b\\in z\\} \\}$ \n",
"4. (Lemma 3) $\\Longleftrightarrow$ $m=\\{a \\mid a\\in x \\vee (a\\in y \\wedge a\\in z) \\}$\n",
"5. (Lemma 4) $\\Longleftrightarrow$ $m=\\{a \\mid (a\\in x \\vee a\\in y) \\wedge (a\\in x \\vee a\\in z) \\}$\n",
"6. (Lemma 3, $\\Leftarrow$) $\\Longleftrightarrow$ $m=\\{a \\mid a\\in \\{b \\mid b\\in x \\vee b\\in y\\} \\wedge a\\in \\{b \\mid b\\in x \\vee b\\in z\\} \\}$\n",
"7. (Lemma 1, $\\Leftarrow$) $\\Longleftrightarrow$ $m=\\{a \\mid a\\in x \\cup y \\wedge a\\in x \\cup z) \\}$\n",
"8. (Lemma 2, $\\Leftarrow$) $m=(x\\cup y) \\cap (x\\cup z)$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Gesetze für $\\cap$, $\\cup$ und $\\setminus$\n",
"\n",
"Für alle Mengen $x$, $y$, $z$ gilt:\n",
"* $x \\cup y = y \\cup x$ (Kommutativ 1)\n",
"* $x \\cap y = y \\cap x$ (Kommutativ 2)\n",
"* $x \\cup (y \\cup z) = (x\\cup y) \\cup z$ (Assoziativ 1)\n",
"* $x \\cap (y \\cap z) = (x\\cap y) \\cap z$ (Assoziativ 2)\n",
"* $x \\cup (y \\cap z) = (x\\cup y) \\cap (x\\cup z)$ (Distributiv 1, siehe oben)\n",
"* $x \\cap (y \\cup z) = (x\\cap y) \\cup (x\\cap z)$ (Distributiv 2)\n",
"* $z \\setminus (x \\cup y) = (z\\setminus x) \\cap (z\\setminus y)$ (De Morgan 1)\n",
"* $z \\setminus (x \\cap y) = (z\\setminus x) \\cup (z\\setminus y)$ (De Morgan 2)\n",
"* $x \\cup \\emptyset = x$ (Leere Menge 1)\n",
"* $x \\cap \\emptyset = \\emptyset$ (Leere Menge 2)\n",
"* $x \\cap (z \\setminus x) = \\emptyset$ (Leere Menge 3)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,3,7,9\\}$"
],
"text/plain": [
"{1,3,7,9}"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(1..10) \\ ({2,4,6,8} ∪ {5,10}) // De Morgan 1 - linke Seite"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,3,5,7,9,10\\}$"
],
"text/plain": [
"{1,3,5,7,9,10}"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(1..10) \\ {2,4,6,8}"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,2,3,4,6,7,8,9\\}$"
],
"text/plain": [
"{1,2,3,4,6,7,8,9}"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(1..10) \\ {5,10}"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,3,7,9\\}$"
],
"text/plain": [
"{1,3,7,9}"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"((1..10) \\ {2,4,6,8}) ∩ ((1..10) \\ {5,10}) // De Morgan 1 - rechte Seite"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Gesetze von De Morgan haben wir auch schon in der Aussagenlogik gesehen.\n",
"Auch hier haben diese die Form wo ein Ausdruck \n",
"* $Op1( x ~ Op2 ~ y)$\n",
"umgewandelt wird in einen Ausdruck\n",
"* $Op1(x) ~ Op2' ~ Op1(y)$\n",
"wo Op2' der duale Operator von Op2 ist."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Hier ein Beispiel für das letzte Gesetz:"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$"
],
"text/plain": [
"∅"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{2,4} ∩ (ℤ \\ {2,4})"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Teilmenge\n",
"\n",
"\n",
"Eine Menge $x$ ist eine Teilmenge oder Untermenge von $y$ wenn gilt:\n",
"* $\\forall a. a\\in x \\Rightarrow a \\in y$\n",
"\n",
"Wir benutzen diese Schreibweise $x \\subseteq y$.\n",
"In diesem Fall ist $y$ auch eine Obermenge von $x$, geschrieben als $y \\supseteq x$.\n",
"\n",
"Für die echte Teilmenge benutzen wir folgende Schreibweise und Definition:\n",
"* $x \\subset y$ $\\Leftrightarrow$ $(x \\subseteq y \\wedge x\\neq y)$."
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\newcommand{\\upto}{\\mathbin{.\\mkern1mu.}}\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{x} = \\{1,3\\}$\n",
"* $\\mathit{y} = (1 \\upto 5)$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
"\tx = {1,3}\n",
"\ty = (1 ‥ 5)"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x={1,3} ∧ y = 1..5 ∧ ∀a.(a∈x ⇒ a∈y)"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{FALSE}$"
],
"text/plain": [
"FALSE"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x={1,3} ∧ y = 1..5 ∧ ∀a.(a∈y ⇒ a∈x)"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$"
],
"text/plain": [
"TRUE"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{1,3} ⊂ 1..5"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Kardinalität\n",
"\n",
"Die Anzahl der Elemente einer Menge $x$ schreiben wir als \n",
"* $\\mid x\\mid$ oder auch als $card(x)$ (B Schreibweise)."
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$3$"
],
"text/plain": [
"3"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card({1,2,3})"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$3$"
],
"text/plain": [
"3"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card({1,1,2,3,2})"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$0$"
],
"text/plain": [
"0"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card(∅)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Achtung, die Kardinalität kann auch unendlich sein: je nach Formalismus, ist folgender Ausdruck entweder unendlich oder nicht wohl definiert: $\\mid \\{x \\mid x>0\\} \\mid$"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"ename": "CommandExecutionException",
"evalue": ":eval: NOT-WELL-DEFINED: \ncard applied to very large set, cardinality not representable in ProB: closure([x],[integer],b(greater(b(identifier(...),integer,[...]),b(value(...),integer,[...])),pred,[nodeid(pos(...))]))\n\n",
"output_type": "error",
"traceback": [
"\u001b[1m\u001b[31m:eval: NOT-WELL-DEFINED: \u001b[0m",
"\u001b[1m\u001b[31mcard applied to very large set, cardinality not representable in ProB: closure([x],[integer],b(greater(b(identifier(...),integer,[...]),b(value(...),integer,[...])),pred,[nodeid(pos(...))]))\u001b[0m"
]
}
],
"source": [
"card({x|x>0})"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# SEND+MORE=MONEY\n",
"\n",
"Als Übung und Beispiel versuchen wir nun ein Problem zu lösen.\n",
"Es handelt sich um ein klassisches arithmetisches Puzzle wo acht unterschiedliche Ziffern gefunden werden sollen die folgende Gleichung erfüllen:\n",
"\n",
"| | | | | |\n",
"|---|---|---|---|---|\n",
"| | S | E | N | D |\n",
"| + | M | O | R | E |\n",
" |\n",
"|= M| O | N | E | Y |\n",
"| | | | | |\n",
"\n",
"Wir können dies nun in Logik, Mengentheorie und Arithmetik modellieren und lösen.\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Wir haben acht Ziffern:"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{R} = 0$\n",
"* $\\mathit{S} = 0$\n",
"* $\\mathit{D} = 0$\n",
"* $\\mathit{E} = 0$\n",
"* $\\mathit{Y} = 0$\n",
"* $\\mathit{M} = 0$\n",
"* $\\mathit{N} = 0$\n",
"* $\\mathit{O} = 0$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
"\tR = 0\n",
"\tS = 0\n",
"\tD = 0\n",
"\tE = 0\n",
"\tY = 0\n",
"\tM = 0\n",
"\tN = 0\n",
"\tO = 0"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{S,E,N,D,M,O,R,Y} ⊆ 0..9"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"diese Ziffern sind alle unterschiedlich:"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{R} = 2$\n",
"* $\\mathit{S} = 0$\n",
"* $\\mathit{D} = 5$\n",
"* $\\mathit{E} = 7$\n",
"* $\\mathit{Y} = 1$\n",
"* $\\mathit{M} = 4$\n",
"* $\\mathit{N} = 6$\n",
"* $\\mathit{O} = 3$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
"\tR = 2\n",
"\tS = 0\n",
"\tD = 5\n",
"\tE = 7\n",
"\tY = 1\n",
"\tM = 4\n",
"\tN = 6\n",
"\tO = 3"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{S,E,N,D,M,O,R,Y} ⊆ 0..9 ∧\n",
"card({S,E,N,D,M,O,R,Y}) = 8"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"und wobei die zwei führenden Ziffern S und M ungleich 0 sind:"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{R} = 3$\n",
"* $\\mathit{S} = 2$\n",
"* $\\mathit{D} = 5$\n",
"* $\\mathit{E} = 7$\n",
"* $\\mathit{Y} = 0$\n",
"* $\\mathit{M} = 1$\n",
"* $\\mathit{N} = 6$\n",
"* $\\mathit{O} = 4$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
"\tR = 3\n",
"\tS = 2\n",
"\tD = 5\n",
"\tE = 7\n",
"\tY = 0\n",
"\tM = 1\n",
"\tN = 6\n",
"\tO = 4"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{S,E,N,D,M,O,R,Y} ⊆ 0..9 ∧ card({S,E,N,D,M,O,R,Y}) = 8 ∧\n",
"S ≠ 0 ∧ M ≠ 0"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"und wo die Summe von SEND+MORE MONEY ergibt:"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{R} = 8$\n",
"* $\\mathit{S} = 9$\n",
"* $\\mathit{D} = 7$\n",
"* $\\mathit{E} = 5$\n",
"* $\\mathit{Y} = 2$\n",
"* $\\mathit{M} = 1$\n",
"* $\\mathit{N} = 6$\n",
"* $\\mathit{O} = 0$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
"\tR = 8\n",
"\tS = 9\n",
"\tD = 7\n",
"\tE = 5\n",
"\tY = 2\n",
"\tM = 1\n",
"\tN = 6\n",
"\tO = 0"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{S,E,N,D,M,O,R,Y} ⊆ 0..9 ∧ card({S,E,N,D,M,O,R,Y}) = 8 ∧ S ≠ 0 & M ≠ 0 ∧\n",
" S*1000 + E*100 + N*10 + D +\n",
" M*1000 + O*100 + R*10 + E =\n",
" M*10000 + O*1000 + N*100 + E*10 + Y"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Aber ist dies die einzige Lösung ? Wir können einfach die Menge aller Lösungen berechnen:"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(9\\mapsto 5\\mapsto 6\\mapsto 7\\mapsto 1\\mapsto 0\\mapsto 8\\mapsto 2)\\}$"
],
"text/plain": [
"{(9↦5↦6↦7↦1↦0↦8↦2)}"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
" {S,E,N,D, M,O,R, Y |\n",
" {S,E,N,D, M,O,R, Y} ⊆ 0..9 ∧ S >0 ∧ M >0 ∧ \n",
" card({S,E,N,D, M,O,R, Y}) = 8 ∧ \n",
" S*1000 + E*100 + N*10 + D +\n",
" M*1000 + O*100 + R*10 + E =\n",
" M*10000 + O*1000 + N*100 + E*10 + Y }"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"|S|E|N|D|M|O|R|Y|\n",
"|---|---|---|---|---|---|---|---|\n",
"|$9$|$5$|$6$|$7$|$1$|$0$|$8$|$2$|\n"
],
"text/plain": [
"S\tE\tN\tD\tM\tO\tR\tY\n",
"9\t5\t6\t7\t1\t0\t8\t2\n"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":table {S,E,N,D, M,O,R, Y |\n",
" {S,E,N,D, M,O,R, Y} ⊆ 0..9 ∧ S >0 ∧ M >0 ∧ \n",
" card({S,E,N,D, M,O,R, Y}) = 8 ∧ \n",
" S*1000 + E*100 + N*10 + D +\n",
" M*1000 + O*100 + R*10 + E =\n",
" M*10000 + O*1000 + N*100 + E*10 + Y }"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"| | | | | |\n",
"|---|---|---|---|---|\n",
"| | S=9 | E=5 | N=6 | D=7 |\n",
"| + | M=1 | O=0 | R=8 | E=5 |\n",
" |\n",
"|= M=1| O=0 | N=6 | E=5 | Y=2 |\n",
"| | | | | |"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Ein anderes arithmetisches Puzzle ist KISS*KISS=PASSION."
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"|K|I|S|P|A|O|N|\n",
"|---|---|---|---|---|---|---|\n",
"|$2$|$0$|$3$|$4$|$1$|$8$|$9$|\n"
],
"text/plain": [
"K\tI\tS\tP\tA\tO\tN\n",
"2\t0\t3\t4\t1\t8\t9\n"
]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":table {K, I, S, P, A, O, N |\n",
" {K,P} ⊆ 1..9 ∧\n",
" {I,S,A,O,N} ⊆ 0..9 ∧\n",
" (1000*K+100*I+10*S+S) * (1000*K+100*I+10*S+S) \n",
" = 1000000*P+100000*A+10000*S+1000*S+100*I+10*O+N &\n",
" card({K, I, S, P, A, O, N}) = 7}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Mengen von Mengen\n",
"\n",
"Mengen können selber auch Mengen beinhalten\n",
"Dies sind alles unterschiedliche Mengen:\n",
"* $\\{\\{2\\},\\{3,4\\}\\}$\n",
"* $\\{\\{2,3\\},\\{4\\}\\}$\n",
"* $\\{\\{2,3\\},\\{3,4\\}\\}$\n",
"* $\\{\\{2,3,4\\}\\}$\n",
"* $\\{2,3,4\\}$ \n",
"\n",
"Wir haben zum Beispiel:\n",
"* $\\mathit{card}(\\{\\{2,3,4\\}\\}) = 1$\n",
"* $\\mathit{card}(\\{\\{2\\},\\{3,4\\}\\}) = 2$\n",
"* $\\mathit{card}(\\{2,3,4\\}) = 3$\n",
"* $\\{2\\} \\in \\{\\{2\\},\\{3,4\\}\\} $\n",
"* $\\{2\\} \\not\\in \\{\\{2,3\\},\\{4\\}\\} $"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$3$"
],
"text/plain": [
"3"
]
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card({2,3,4})"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$1$"
],
"text/plain": [
"1"
]
},
"execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card({{2,3,4}})"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$2$"
],
"text/plain": [
"2"
]
},
"execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card({{2},{3,4}})"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$"
],
"text/plain": [
"TRUE"
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{2} ∈ {{2}, {3,4}}"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{FALSE}$"
],
"text/plain": [
"FALSE"
]
},
"execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{2} ∈ {{2,3}, {4}}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Mengen von Mengen und die leere Menge\n",
"\n",
"Achtung: man muss die leere Menge von der Menge die die leere Menge beinhaltet unterscheiden:\n",
"* $\\emptyset \\neq \\{\\emptyset\\}$"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{FALSE}$"
],
"text/plain": [
"FALSE"
]
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"∅ = {∅}"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$0$"
],
"text/plain": [
"0"
]
},
"execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card(∅)"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$1$"
],
"text/plain": [
"1"
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card({∅})"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$"
],
"text/plain": [
"TRUE"
]
},
"execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"∅ ∈ {∅}"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{FALSE}$"
],
"text/plain": [
"FALSE"
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"∅ ∈ ∅"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$"
],
"text/plain": [
"TRUE"
]
},
"execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"∅ ∉ ∅"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Potenzmenge\n",
"Die Menge aller Untermengen einer Menge $A$ schreiben wir als $ℙ(\\mathit{A}) $ oder auch als $2^{A}$."
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\{\\emptyset,\\{1\\},\\{1,2\\},\\{2\\}\\}$"
],
"text/plain": [
"{∅,{1},{1,2},{2}}"
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ℙ({1,2})"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\{\\emptyset\\}$"
],
"text/plain": [
"{∅}"
]
},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ℙ(∅)"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\{\\emptyset,\\{1\\},\\{1,2\\},\\{1,3\\},\\{2\\},\\{1,2,3\\},\\{2,3\\},\\{3\\}\\}$"
],
"text/plain": [
"{∅,{1},{1,2},{1,3},{2},{1,2,3},{2,3},{3}}"
]
},
"execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ℙ(1..3)"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"|Elements|\n",
"|---|\n",
"|$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$|\n",
"|$\\{1\\}$|\n",
"|$\\{1,2\\}$|\n",
"|$\\{1,3\\}$|\n",
"|$\\{2\\}$|\n",
"|$\\{1,2,3\\}$|\n",
"|$\\{2,3\\}$|\n",
"|$\\{3\\}$|\n"
],
"text/plain": [
"Elements\n",
"{}\n",
"{1}\n",
"{1,2}\n",
"{1,3}\n",
"{2}\n",
"{1,2,3}\n",
"{2,3}\n",
"{3}\n"
]
},
"execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":table ℙ(1..3)"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"|Elements|\n",
"|---|\n",
"|$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$|\n",
"|$\\{\\emptyset\\}$|\n",
"|$\\{\\emptyset,\\{1\\}\\}$|\n",
"|$\\{\\{1\\}\\}$|\n"
],
"text/plain": [
"Elements\n",
"{}\n",
"{{}}\n",
"{{},{1}}\n",
"{{1}}\n"
]
},
"execution_count": 54,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":table ℙ(ℙ({1}))"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$8$"
],
"text/plain": [
"8"
]
},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card(ℙ(1..3))"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$1024$"
],
"text/plain": [
"1024"
]
},
"execution_count": 56,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card(ℙ(1..10))"
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$1267650600228229401496703205376$"
],
"text/plain": [
"1267650600228229401496703205376"
]
},
"execution_count": 57,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card(ℙ(1..100))"
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$340282366920938463463374607431768211456$"
],
"text/plain": [
"340282366920938463463374607431768211456"
]
},
"execution_count": 58,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card(ℙ(ℙ(1..7)))"
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$115792089237316195423570985008687907853269984665640564039457584007913129639936$"
],
"text/plain": [
"115792089237316195423570985008687907853269984665640564039457584007913129639936"
]
},
"execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card(ℙ(ℙ(ℙ(1..3))))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Relationen in der Mengentheorie\n",
"* Was ist eine Relation?\n",
"* Wie kann man Relationen in Mengentheorie und Logik abbilden?\n",
"\n",
"* https://en.wikipedia.org/wiki/Finitary\\_relation}:\n",
" * _In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple._\n",
"\n",
"* k-Tupel: $(1,2)$ (k=2, Paar)\n",
"* Eine Relation weist k-Tupeln Wahrheitswerte zu: \n",
" * Beispiel: $(1,2) \\mapsto TRUE$, $(2,1) \\mapsto FALSE$.\n",
"\n",
"## Unäre Relationen\n",
"\n",
"Eine unäre Relation über $x$ entspricht einfach einer Untermenge von $x$:\n",
"* die Menge an Werten für die die Relation wahr ist.\n",
"\n",
"Beispiele:\n",
"* Relation ``Ziffer'' über die ganzen Zahlen ist\n",
" * $0 .. 9 \\subseteq$ ℤ\n",
"* Relation ``Gt0'' über die ganzen Zahlen ist\n",
" * $\\{ \\mathit{x}|\\mathit{x} \\in $ℤ $\\wedge \\mathit{x} > 0\\} \\subseteq$ ℤ.\n",
"* Relation ``Gerade'' über die ganzen Zahlen ist\n",
" * $\\{ \\mathit{x}|\\mathit{x} \\in $ ℤ $\\wedge \\mathit{x} \\mod 2 = 0\\} \\subseteq$ ℤ.\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"## Relationen vs Prädikate\n",
"Eine Relation ist die explizite Darstellung eines Prädikats als Menge:\n",
" \n",
"* Prädikate können mit logischen Junktoren und Quantoren verarbeitet werden:\n",
" * $\\exists x. is\\_rich(x)$, $\\forall x.(is\\_poor(x) \\Rightarrow \\neg is\\_rich(x))$\n",
" \n",
"* Relationen können mit mengentheoretischen Operationen verarbeitet werden:\n",
" * $rich \\neq \\emptyset$, $rich \\cap poor = \\emptyset$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Kartesisches Produkt und Paare\n",
"\n",
"Das kartesische Produkt $x \\times y$ zweier Mengen $x$ und $y$ ist definiert als\n",
"* $\\{ (a,b) \\mid a\\in x \\wedge b\\in y\\}$.\n",
"\n",
"$(a,b)$ steht hier für ein geordnetes Paar,\n",
" d.h. $(1 \\mapsto 2) \\neq (2 \\mapsto 1) $. \n",
"Wir schreiben manchmal auch $a \\mapsto b$ anstatt $(a,b)$."
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto 4),(1\\mapsto 5),(2\\mapsto 4),(2\\mapsto 5)\\}$"
],
"text/plain": [
"{(1↦4),(1↦5),(2↦4),(2↦5)}"
]
},
"execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(1 .. 2) × (4 .. 5)"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto 4),(2\\mapsto 4)\\}$"
],
"text/plain": [
"{(1↦4),(2↦4)}"
]
},
"execution_count": 61,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(1 .. 2) × {4}"
]
},
{
"cell_type": "code",
"execution_count": 62,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto \\mathit{FALSE}),(1\\mapsto \\mathit{TRUE}),(2\\mapsto \\mathit{FALSE}),(2\\mapsto \\mathit{TRUE})\\}$"
],
"text/plain": [
"{(1↦FALSE),(1↦TRUE),(2↦FALSE),(2↦TRUE)}"
]
},
"execution_count": 62,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(1 .. 2) × BOOL"
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto 1),(1\\mapsto 2),(1\\mapsto 3),(2\\mapsto 1),(2\\mapsto 2),(2\\mapsto 3),(3\\mapsto 1),(3\\mapsto 2),(3\\mapsto 3)\\}$"
],
"text/plain": [
"{(1↦1),(1↦2),(1↦3),(2↦1),(2↦2),(2↦3),(3↦1),(3↦2),(3↦3)}"
]
},
"execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(1 .. 3) × (1 .. 3)"
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$100$"
],
"text/plain": [
"100"
]
},
"execution_count": 64,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card((1..10)×(1..10))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Das Kartesische Produkt ist nicht kommutativ:"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(2\\mapsto 3)\\}$"
],
"text/plain": [
"{(2↦3)}"
]
},
"execution_count": 65,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{2}×{3}"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(3\\mapsto 2)\\}$"
],
"text/plain": [
"{(3↦2)}"
]
},
"execution_count": 66,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{3}×{2}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Es gilt:\n",
"* $A \\times B = \\emptyset \\equiv (A = \\emptyset \\vee B=\\emptyset)$.\n",
"Das Kartesische Produkt wird auch Kreuzmenge oder Produktmenge genannt."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Binäre Relationen\n",
"\n",
"Eine binäre Relation über $x$ und $y$ ist eine Untermenge des kartesischen Produkts $x \\times y$ zweier Mengen. Die Mengen $x$ und $y$ können identisch sein.\n",
"\n",
"Beispiel: die Relation ``kleiner'' über die Ziffern $0..9$:\n"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(0\\mapsto 1),(0\\mapsto 2),(0\\mapsto 3),(0\\mapsto 4),(0\\mapsto 5),(0\\mapsto 6),(0\\mapsto 7),(0\\mapsto 8),(0\\mapsto 9),(1\\mapsto 2),(1\\mapsto 3),(1\\mapsto 4),(1\\mapsto 5),(1\\mapsto 6),(1\\mapsto 7),(1\\mapsto 8),(1\\mapsto 9),(2\\mapsto 3),(2\\mapsto 4),(2\\mapsto 5),(2\\mapsto 6),(2\\mapsto 7),(2\\mapsto 8),(2\\mapsto 9),(3\\mapsto 4),(3\\mapsto 5),(3\\mapsto 6),(3\\mapsto 7),(3\\mapsto 8),(3\\mapsto 9),(4\\mapsto 5),(4\\mapsto 6),(4\\mapsto 7),(4\\mapsto 8),(4\\mapsto 9),(5\\mapsto 6),(5\\mapsto 7),(5\\mapsto 8),(5\\mapsto 9),(6\\mapsto 7),(6\\mapsto 8),(6\\mapsto 9),(7\\mapsto 8),(7\\mapsto 9),(8\\mapsto 9)\\}$"
],
"text/plain": [
"{(0↦1),(0↦2),(0↦3),(0↦4),(0↦5),(0↦6),(0↦7),(0↦8),(0↦9),(1↦2),(1↦3),(1↦4),(1↦5),(1↦6),(1↦7),(1↦8),(1↦9),(2↦3),(2↦4),(2↦5),(2↦6),(2↦7),(2↦8),(2↦9),(3↦4),(3↦5),(3↦6),(3↦7),(3↦8),(3↦9),(4↦5),(4↦6),(4↦7),(4↦8),(4↦9),(5↦6),(5↦7),(5↦8),(5↦9),(6↦7),(6↦8),(6↦9),(7↦8),(7↦9),(8↦9)}"
]
},
"execution_count": 67,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{a,b| (a,b)∈(0..9)×(0..9) ∧ a<b}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Diese Relation ist eine Untermenge von ℤ×ℤ.\n",
"Ein anderes Beispiel ist die Relation ``halb'' über die ganzen Zahlen $1..10$:"
]
},
{
"cell_type": "code",
"execution_count": 68,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(2\\mapsto 1),(4\\mapsto 2),(6\\mapsto 3),(8\\mapsto 4),(10\\mapsto 5)\\}$"
],
"text/plain": [
"{(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)}"
]
},
"execution_count": 68,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{a,b| a∈1..10 ∧ b∈1..10 ∧ b*2=a}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Eine binäre Relation kann auch als gerichteter Graph angesehen werden:"
]
},
{
"cell_type": "code",
"execution_count": 69,
"metadata": {},
"outputs": [
{
"data": {
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"<g id=\"node5\" class=\"node\">\n",
"<title>6</title>\n",
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"</g>\n",
"<!-- 3 -->\n",
"<g id=\"node6\" class=\"node\">\n",
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"</g>\n",
"<!-- 6->3 -->\n",
"<g id=\"edge3\" class=\"edge\">\n",
"<title>6->3</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"174.5,-220.18 171,-210.18 167.5,-220.18 174.5,-220.18\"/>\n",
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"<text text-anchor=\"middle\" x=\"99\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
"</g>\n",
"<!-- 2->1 -->\n",
"<g id=\"edge5\" class=\"edge\">\n",
"<title>2->1</title>\n",
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"</g>\n",
"</svg>"
],
"text/plain": [
"<Dot visualization: expr_as_graph [(\"halb\",{a,b|a:1..10 & b:1..10 & b*2=a})]>"
]
},
"execution_count": 69,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":dot expr_as_graph (\"halb\",{a,b| a∈1..10 ∧ b∈1..10 & b*2=a})"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Die Typen der Elemente kann abweichen.\n",
"Zum Beispiel die Relation ``durch drei teilbar'' über die ganzen Zahlen $1..7$ zum Datentypen $STRING$"
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto\\text{\"nein\"}),(2\\mapsto\\text{\"nein\"}),(3\\mapsto\\text{\"ja\"}),(4\\mapsto\\text{\"nein\"}),(5\\mapsto\\text{\"nein\"}),(6\\mapsto\\text{\"ja\"}),(7\\mapsto\\text{\"nein\"})\\}$"
],
"text/plain": [
"{(1↦\"nein\"),(2↦\"nein\"),(3↦\"ja\"),(4↦\"nein\"),(5↦\"nein\"),(6↦\"ja\"),(7↦\"nein\")}"
]
},
"execution_count": 70,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{x,y| x:1..7 ∧ (x mod 3 =0 => y = \"ja\") ∧ (x mod 3 >0 => y= \"nein\")}"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {},
"outputs": [
{
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"text/plain": [
"<Dot visualization: expr_as_graph [(\"teilbar\",{x,y|x:1..7 & (x mod 3=0 => y=\"ja\") & (x mod 3>0 => y=\"nein\")})]>"
]
},
"execution_count": 71,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":dot expr_as_graph (\"teilbar\",{x,y| x:1..7 ∧ (x mod 3 =0 => y = \"ja\") ∧ (x mod 3 >0 => y= \"nein\")})"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Das Kartesische Produkt stellt die Relation dar, die immer wahr ist (für die angegebenen Basismengen). Als Graph ist dies der [vollständige Graph](https://de.wikipedia.org/wiki/Vollständiger_Graph) über die Basismengen:"
]
},
{
"cell_type": "code",
"execution_count": 72,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Preference changed: DOT_ENGINE = circo\n"
]
},
"execution_count": 72,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":pref DOT_ENGINE=circo"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {},
"outputs": [
{
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"text/plain": [
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]
},
"execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":dot expr_as_graph (\"K5\",(1..5)×(1..5))"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Preference changed: DOT_ENGINE = circo\n"
]
},
"execution_count": 74,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":pref DOT_ENGINE=circo"
]
},
{
"cell_type": "code",
"execution_count": 75,
"metadata": {},
"outputs": [
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"<title>8->2</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"363.12,-305.81 372.14,-311.36 368.55,-301.4 363.12,-305.81\"/>\n",
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"<title>8->1</title>\n",
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"<title>7->10</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"352.72,-101.81 362.51,-97.78 352.34,-94.82 352.72,-101.81\"/>\n",
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"<title>7->9</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"390.48,-207.58 401.05,-206.8 392.3,-200.82 390.48,-207.58\"/>\n",
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"<title>7->8</title>\n",
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"<title>7->7</title>\n",
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"<title>7->6</title>\n",
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"<title>7->5</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"57.25,-301.69 61.49,-311.4 64.23,-301.16 57.25,-301.69\"/>\n",
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"<title>7->4</title>\n",
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"<title>7->3</title>\n",
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"<title>7->2</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"352.38,-312.01 362.66,-314.58 356.23,-306.16 352.38,-312.01\"/>\n",
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"<title>7->1</title>\n",
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"<title>6->10</title>\n",
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"<title>6->8</title>\n",
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"<title>6->7</title>\n",
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"<title>6->6</title>\n",
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"<title>6->5</title>\n",
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"<title>6->4</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"137.79,-378.64 146.84,-384.14 143.19,-374.2 137.79,-378.64\"/>\n",
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"<title>6->3</title>\n",
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"</g>\n",
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"<title>6->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M54.28,-224.6C118.69,-249.47 280.18,-302.07 353.07,-322.07\"/>\n",
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"</g>\n",
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"<title>6->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M54.13,-197.11C104.84,-164.44 213.16,-85.85 263.22,-45.53\"/>\n",
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"<title>5->10</title>\n",
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"<title>5->9</title>\n",
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"<title>5->8</title>\n",
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"<title>5->7</title>\n",
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"<title>5->6</title>\n",
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"<title>5->5</title>\n",
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"<title>5->4</title>\n",
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"<title>5->3</title>\n",
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"<title>5->2</title>\n",
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"<title>5->1</title>\n",
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"<title>4->9</title>\n",
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"<title>4->8</title>\n",
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"<title>4->7</title>\n",
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"<title>4->4</title>\n",
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"text/plain": [
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},
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"metadata": {},
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}
],
"source": [
":dot expr_as_graph (\"K10\",(1..10)×(1..10))"
]
},
{
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"</g>\n",
"<!-- 17 -->\n",
"<g id=\"node4\" class=\"node\">\n",
"<title>17</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"579.01,-55.63 525.01,-55.63 525.01,-19.63 579.01,-19.63 579.01,-55.63\"/>\n",
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"</g>\n",
"<!-- 20->17 -->\n",
"<g id=\"edge4\" class=\"edge\">\n",
"<title>20->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M796.23,-276.97C756.75,-231.77 640.09,-114.65 583.45,-62.83\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"585.63,-60.08 575.87,-55.96 580.94,-65.27 585.63,-60.08\"/>\n",
"<text text-anchor=\"middle\" x=\"675.34\" y=\"-173.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16 -->\n",
"<g id=\"node5\" class=\"node\">\n",
"<title>16</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"455.07,-36 401.07,-36 401.07,0 455.07,0 455.07,-36\"/>\n",
"<text text-anchor=\"middle\" x=\"428.07\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">16</text>\n",
"</g>\n",
"<!-- 20->16 -->\n",
"<g id=\"edge5\" class=\"edge\">\n",
"<title>20->16</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M790.14,-276.95C729.46,-227.52 542.82,-91.63 463.87,-39.03\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.51,-35.92 455.24,-33.33 461.65,-41.76 465.51,-35.92\"/>\n",
"<text text-anchor=\"middle\" x=\"612.5\" y=\"-161.79\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15 -->\n",
"<g id=\"node6\" class=\"node\">\n",
"<title>15</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"331.13,-55.63 277.13,-55.63 277.13,-19.63 331.13,-19.63 331.13,-55.63\"/>\n",
"<text text-anchor=\"middle\" x=\"304.13\" y=\"-33.93\" font-family=\"Times,serif\" font-size=\"14.00\">15</text>\n",
"</g>\n",
"<!-- 20->15 -->\n",
"<g id=\"edge6\" class=\"edge\">\n",
"<title>20->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M782.41,-277.58C697.52,-229.33 436.9,-96.43 340.48,-52.07\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.69,-48.78 331.14,-47.82 338.79,-55.15 341.69,-48.78\"/>\n",
"<text text-anchor=\"middle\" x=\"546.94\" y=\"-153.63\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14 -->\n",
"<g id=\"node7\" class=\"node\">\n",
"<title>14</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"219.33,-112.6 165.33,-112.6 165.33,-76.6 219.33,-76.6 219.33,-112.6\"/>\n",
"<text text-anchor=\"middle\" x=\"192.33\" y=\"-90.9\" font-family=\"Times,serif\" font-size=\"14.00\">14</text>\n",
"</g>\n",
"<!-- 20->14 -->\n",
"<g id=\"edge7\" class=\"edge\">\n",
"<title>20->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M782.33,-283.11C683.43,-245.89 342.78,-135.09 229.37,-103.08\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"230,-99.62 219.43,-100.32 228.13,-106.37 230,-99.62\"/>\n",
"<text text-anchor=\"middle\" x=\"491.35\" y=\"-181.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 13 -->\n",
"<g id=\"node8\" class=\"node\">\n",
"<title>13</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"130.6,-201.33 76.6,-201.33 76.6,-165.33 130.6,-165.33 130.6,-201.33\"/>\n",
"<text text-anchor=\"middle\" x=\"103.6\" y=\"-179.63\" font-family=\"Times,serif\" font-size=\"14.00\">13</text>\n",
"</g>\n",
"<!-- 20->13 -->\n",
"<g id=\"edge8\" class=\"edge\">\n",
"<title>20->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M782.28,-287.95C673.1,-265.51 266.92,-201.04 141.04,-185.99\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"141.09,-182.48 130.76,-184.8 140.29,-189.43 141.09,-182.48\"/>\n",
"<text text-anchor=\"middle\" x=\"447.16\" y=\"-225.77\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 12 -->\n",
"<g id=\"node9\" class=\"node\">\n",
"<title>12</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"73.63,-313.13 19.63,-313.13 19.63,-277.13 73.63,-277.13 73.63,-313.13\"/>\n",
"<text text-anchor=\"middle\" x=\"46.63\" y=\"-291.43\" font-family=\"Times,serif\" font-size=\"14.00\">12</text>\n",
"</g>\n",
"<!-- 20->12 -->\n",
"<g id=\"edge9\" class=\"edge\">\n",
"<title>20->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M782.36,-292.42C666.97,-287.18 217.12,-287.04 83.96,-292\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"83.7,-288.51 73.85,-292.42 83.99,-295.5 83.7,-288.51\"/>\n",
"<text text-anchor=\"middle\" x=\"418.66\" y=\"-281.01\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 11 -->\n",
"<g id=\"node10\" class=\"node\">\n",
"<title>11</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-437.07 0,-437.07 0,-401.07 54,-401.07 54,-437.07\"/>\n",
"<text text-anchor=\"middle\" x=\"27\" y=\"-415.37\" font-family=\"Times,serif\" font-size=\"14.00\">11</text>\n",
"</g>\n",
"<!-- 20->11 -->\n",
"<g id=\"edge10\" class=\"edge\">\n",
"<title>20->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M782.16,-296.76C664.25,-310.08 199.36,-383.57 64.1,-410.05\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"63.33,-406.64 54.21,-412.02 64.7,-413.5 63.33,-406.64\"/>\n",
"<text text-anchor=\"middle\" x=\"408.63\" y=\"-342.21\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 10 -->\n",
"<g id=\"node11\" class=\"node\">\n",
"<title>10</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"73.63,-561.01 19.63,-561.01 19.63,-525.01 73.63,-525.01 73.63,-561.01\"/>\n",
"<text text-anchor=\"middle\" x=\"46.63\" y=\"-539.31\" font-family=\"Times,serif\" font-size=\"14.00\">10</text>\n",
"</g>\n",
"<!-- 20->10 -->\n",
"<g id=\"edge11\" class=\"edge\">\n",
"<title>20->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M782.43,-301.12C667.04,-333.07 216.43,-479.31 83.64,-527.67\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"82.1,-524.5 73.92,-531.24 84.52,-531.07 82.1,-524.5\"/>\n",
"<text text-anchor=\"middle\" x=\"447.53\" y=\"-403.2\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 9 -->\n",
"<g id=\"node12\" class=\"node\">\n",
"<title>9</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"130.6,-672.81 76.6,-672.81 76.6,-636.81 130.6,-636.81 130.6,-672.81\"/>\n",
"<text text-anchor=\"middle\" x=\"103.6\" y=\"-651.11\" font-family=\"Times,serif\" font-size=\"14.00\">9</text>\n",
"</g>\n",
"<!-- 20->9 -->\n",
"<g id=\"edge12\" class=\"edge\">\n",
"<title>20->9</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M782.3,-305.91C672.7,-355.96 263.74,-564.15 139.56,-632.85\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"137.82,-629.82 130.78,-637.74 141.22,-635.93 137.82,-629.82\"/>\n",
"<text text-anchor=\"middle\" x=\"446.43\" y=\"-458.18\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8 -->\n",
"<g id=\"node13\" class=\"node\">\n",
"<title>8</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"219.33,-761.54 165.33,-761.54 165.33,-725.54 219.33,-725.54 219.33,-761.54\"/>\n",
"<text text-anchor=\"middle\" x=\"192.33\" y=\"-739.84\" font-family=\"Times,serif\" font-size=\"14.00\">8</text>\n",
"</g>\n",
"<!-- 20->8 -->\n",
"<g id=\"edge13\" class=\"edge\">\n",
"<title>20->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M782.47,-311.21C681.09,-378.6 324.52,-637.57 220.8,-718.98\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"218.33,-716.47 212.66,-725.42 222.67,-721.96 218.33,-716.47\"/>\n",
"<text text-anchor=\"middle\" x=\"487.13\" y=\"-503.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 7 -->\n",
"<g id=\"node14\" class=\"node\">\n",
"<title>7</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"331.13,-818.51 277.13,-818.51 277.13,-782.51 331.13,-782.51 331.13,-818.51\"/>\n",
"<text text-anchor=\"middle\" x=\"304.13\" y=\"-796.81\" font-family=\"Times,serif\" font-size=\"14.00\">7</text>\n",
"</g>\n",
"<!-- 20->7 -->\n",
"<g id=\"edge14\" class=\"edge\">\n",
"<title>20->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M787.37,-313.24C703.64,-389.72 407.6,-685.66 325.39,-774.94\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"322.69,-772.7 318.54,-782.44 327.87,-777.41 322.69,-772.7\"/>\n",
"<text text-anchor=\"middle\" x=\"541.88\" y=\"-547.89\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 6 -->\n",
"<g id=\"node15\" class=\"node\">\n",
"<title>6</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"455.07,-838.14 401.07,-838.14 401.07,-802.14 455.07,-802.14 455.07,-838.14\"/>\n",
"<text text-anchor=\"middle\" x=\"428.07\" y=\"-816.44\" font-family=\"Times,serif\" font-size=\"14.00\">6</text>\n",
"</g>\n",
"<!-- 20->6 -->\n",
"<g id=\"edge15\" class=\"edge\">\n",
"<title>20->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M792.76,-313.43C729.31,-391.96 505.3,-700.12 443.73,-793.33\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"440.66,-791.62 438.13,-801.91 446.53,-795.45 440.66,-791.62\"/>\n",
"<text text-anchor=\"middle\" x=\"603.75\" y=\"-557.18\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 5 -->\n",
"<g id=\"node16\" class=\"node\">\n",
"<title>5</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"579.01,-818.51 525.01,-818.51 525.01,-782.51 579.01,-782.51 579.01,-818.51\"/>\n",
"<text text-anchor=\"middle\" x=\"552.01\" y=\"-796.81\" font-family=\"Times,serif\" font-size=\"14.00\">5</text>\n",
"</g>\n",
"<!-- 20->5 -->\n",
"<g id=\"edge16\" class=\"edge\">\n",
"<title>20->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M797.07,-313.36C752.55,-389.45 603.47,-681.77 562.38,-773.1\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"559.07,-771.94 558.23,-782.5 565.47,-774.77 559.07,-771.94\"/>\n",
"<text text-anchor=\"middle\" x=\"665.23\" y=\"-547.03\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 4 -->\n",
"<g id=\"node17\" class=\"node\">\n",
"<title>4</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"690.81,-761.54 636.81,-761.54 636.81,-725.54 690.81,-725.54 690.81,-761.54\"/>\n",
"<text text-anchor=\"middle\" x=\"663.81\" y=\"-739.84\" font-family=\"Times,serif\" font-size=\"14.00\">4</text>\n",
"</g>\n",
"<!-- 20->4 -->\n",
"<g id=\"edge17\" class=\"edge\">\n",
"<title>20->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M800.55,-313.21C772.87,-382.5 691.82,-631.4 669.31,-715.54\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"665.87,-714.84 666.75,-725.4 672.65,-716.6 665.87,-714.84\"/>\n",
"<text text-anchor=\"middle\" x=\"720.43\" y=\"-518.17\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3 -->\n",
"<g id=\"node18\" class=\"node\">\n",
"<title>3</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"779.54,-672.81 725.54,-672.81 725.54,-636.81 779.54,-636.81 779.54,-672.81\"/>\n",
"<text text-anchor=\"middle\" x=\"752.54\" y=\"-651.11\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
"</g>\n",
"<!-- 20->3 -->\n",
"<g id=\"edge18\" class=\"edge\">\n",
"<title>20->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M803.34,-313.46C789.5,-372.15 760.22,-555.72 753.16,-626.55\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"749.67,-626.26 752.26,-636.53 756.65,-626.89 749.67,-626.26\"/>\n",
"<text text-anchor=\"middle\" x=\"763.75\" y=\"-473.8\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 2 -->\n",
"<g id=\"node19\" class=\"node\">\n",
"<title>2</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"836.51,-561.01 782.51,-561.01 782.51,-525.01 836.51,-525.01 836.51,-561.01\"/>\n",
"<text text-anchor=\"middle\" x=\"809.51\" y=\"-539.31\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
"</g>\n",
"<!-- 20->2 -->\n",
"<g id=\"edge19\" class=\"edge\">\n",
"<title>20->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M805.77,-313.36C802.02,-356.16 801.78,-463.32 805.02,-514.68\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"801.54,-515.04 805.77,-524.75 808.52,-514.52 801.54,-515.04\"/>\n",
"<text text-anchor=\"middle\" x=\"790.9\" y=\"-402.82\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g id=\"node20\" class=\"node\">\n",
"<title>1</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"779.54,-201.33 725.54,-201.33 725.54,-165.33 779.54,-165.33 779.54,-201.33\"/>\n",
"<text text-anchor=\"middle\" x=\"752.54\" y=\"-179.63\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
"</g>\n",
"<!-- 20->1 -->\n",
"<g id=\"edge20\" class=\"edge\">\n",
"<title>20->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M805.72,-276.89C798.71,-258.73 784.87,-230.73 772.94,-210.07\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"775.92,-208.24 767.8,-201.44 769.91,-211.82 775.92,-208.24\"/>\n",
"<text text-anchor=\"middle\" x=\"774.83\" y=\"-247.28\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->20 -->\n",
"<g id=\"edge21\" class=\"edge\">\n",
"<title>19->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M831.18,-400.81C830.11,-380.37 825.14,-347 819.85,-323.23\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"823.22,-322.29 817.49,-313.38 816.42,-323.92 823.22,-322.29\"/>\n",
"<text text-anchor=\"middle\" x=\"840.02\" y=\"-365.82\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->19 -->\n",
"<g id=\"edge22\" class=\"edge\">\n",
"<title>19->19</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M856.38,-425.81C866.16,-425.96 874.14,-423.71 874.14,-419.07 874.14,-416.24 871.18,-414.3 866.65,-413.25\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"866.66,-409.74 856.38,-412.33 866.03,-416.71 866.66,-409.74\"/>\n",
"<text text-anchor=\"middle\" x=\"888.64\" y=\"-415.37\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->18 -->\n",
"<g id=\"edge23\" class=\"edge\">\n",
"<title>19->18</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M823.36,-400.62C800.5,-346.04 719,-185.48 682.01,-121.41\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"684.96,-119.53 676.89,-112.68 678.93,-123.08 684.96,-119.53\"/>\n",
"<text text-anchor=\"middle\" x=\"738.18\" y=\"-264.81\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->17 -->\n",
"<g id=\"edge24\" class=\"edge\">\n",
"<title>19->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M819.62,-400.94C780.58,-339.08 633.76,-136.54 575.7,-63.84\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"578.08,-61.22 569.07,-55.64 572.64,-65.62 578.08,-61.22\"/>\n",
"<text text-anchor=\"middle\" x=\"683.16\" y=\"-236.19\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->16 -->\n",
"<g id=\"edge25\" class=\"edge\">\n",
"<title>19->16</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M814.95,-400.82C757.14,-336.08 539.45,-118.03 458.53,-43.3\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"460.43,-40.29 450.69,-36.12 455.7,-45.45 460.43,-40.29\"/>\n",
"<text text-anchor=\"middle\" x=\"622.24\" y=\"-225.86\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->15 -->\n",
"<g id=\"edge26\" class=\"edge\">\n",
"<title>19->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M809.28,-401.07C730.71,-337.89 441.39,-127.44 339.71,-59.13\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.57,-56.17 331.31,-53.53 337.69,-61.99 341.57,-56.17\"/>\n",
"<text text-anchor=\"middle\" x=\"560\" y=\"-233.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->14 -->\n",
"<g id=\"edge27\" class=\"edge\">\n",
"<title>19->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M801.91,-401.81C700.43,-344.62 344.41,-163.12 228.84,-109.5\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"230.22,-106.29 219.67,-105.29 227.3,-112.65 230.22,-106.29\"/>\n",
"<text text-anchor=\"middle\" x=\"500.87\" y=\"-244.46\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->13 -->\n",
"<g id=\"edge28\" class=\"edge\">\n",
"<title>19->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M802.01,-407.3C690.64,-365.71 268.48,-228.42 140.55,-192.02\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"141.39,-188.62 130.82,-189.29 139.5,-195.36 141.39,-188.62\"/>\n",
"<text text-anchor=\"middle\" x=\"456.78\" y=\"-288.46\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->12 -->\n",
"<g id=\"edge29\" class=\"edge\">\n",
"<title>19->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M801.72,-411.98C683.78,-387.98 219.67,-314.35 84.04,-297.94\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"84.09,-294.42 73.76,-296.74 83.28,-301.38 84.09,-294.42\"/>\n",
"<text text-anchor=\"middle\" x=\"457.38\" y=\"-343.76\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->11 -->\n",
"<g id=\"edge30\" class=\"edge\">\n",
"<title>19->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M801.99,-416.42C682.37,-411.09 202.14,-410.96 64.21,-416.01\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"63.99,-412.52 54.14,-416.42 64.27,-419.51 63.99,-412.52\"/>\n",
"<text text-anchor=\"middle\" x=\"418.6\" y=\"-420.02\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->10 -->\n",
"<g id=\"edge31\" class=\"edge\">\n",
"<title>19->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M801.79,-420.7C683.88,-434.02 218.99,-507.5 83.73,-533.99\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"82.96,-530.57 73.84,-535.96 84.33,-537.44 82.96,-530.57\"/>\n",
"<text text-anchor=\"middle\" x=\"428.26\" y=\"-481.14\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->9 -->\n",
"<g id=\"edge32\" class=\"edge\">\n",
"<title>19->9</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M802.05,-425C690.74,-455.71 268.64,-592.69 140.62,-639.39\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"139.05,-636.24 130.88,-642.98 141.47,-642.81 139.05,-636.24\"/>\n",
"<text text-anchor=\"middle\" x=\"456.83\" y=\"-521\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->8 -->\n",
"<g id=\"edge33\" class=\"edge\">\n",
"<title>19->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M802.08,-429.63C700.7,-475.69 343.48,-657.51 228.35,-721.37\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"226.56,-718.36 219.54,-726.3 229.98,-724.47 226.56,-718.36\"/>\n",
"<text text-anchor=\"middle\" x=\"500.72\" y=\"-564.3\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->7 -->\n",
"<g id=\"edge34\" class=\"edge\">\n",
"<title>19->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M801.99,-434.95C711.89,-494.52 423.64,-703.84 332.2,-775.97\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"329.88,-773.35 324.23,-782.32 334.23,-778.83 329.88,-773.35\"/>\n",
"<text text-anchor=\"middle\" x=\"552.6\" y=\"-594.26\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->6 -->\n",
"<g id=\"edge35\" class=\"edge\">\n",
"<title>19->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M806.45,-437.26C735.37,-501.72 518.25,-718.74 449.27,-794.14\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"446.45,-792.04 442.33,-801.81 451.64,-796.74 446.45,-792.04\"/>\n",
"<text text-anchor=\"middle\" x=\"613.36\" y=\"-619.5\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->5 -->\n",
"<g id=\"edge36\" class=\"edge\">\n",
"<title>19->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M812.02,-437.15C761.64,-498.69 615.33,-699.87 567.33,-773.37\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"564.19,-771.77 561.72,-782.08 570.08,-775.56 564.19,-771.77\"/>\n",
"<text text-anchor=\"middle\" x=\"675.17\" y=\"-609.06\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->4 -->\n",
"<g id=\"edge37\" class=\"edge\">\n",
"<title>19->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M815.97,-437.31C783.64,-491.41 702.18,-650.97 673.62,-715.76\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"670.24,-714.75 669.5,-725.32 676.67,-717.52 670.24,-714.75\"/>\n",
"<text text-anchor=\"middle\" x=\"730.29\" y=\"-580.34\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->3 -->\n",
"<g id=\"edge38\" class=\"edge\">\n",
"<title>19->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M819.15,-437.15C802.09,-478.05 769.49,-577.79 756.96,-626.77\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"753.51,-626.14 754.54,-636.69 760.31,-627.8 753.51,-626.14\"/>\n",
"<text text-anchor=\"middle\" x=\"773.55\" y=\"-520.76\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->2 -->\n",
"<g id=\"edge39\" class=\"edge\">\n",
"<title>19->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M821.21,-437.11C815.91,-457.46 810.39,-490.81 808.21,-514.65\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"804.71,-514.62 807.46,-524.85 811.69,-515.13 804.71,-514.62\"/>\n",
"<text text-anchor=\"middle\" x=\"800.21\" y=\"-479.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 19->1 -->\n",
"<g id=\"edge40\" class=\"edge\">\n",
"<title>19->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M827.15,-400.97C817.74,-359.92 785.46,-259.6 766.33,-210.83\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"769.56,-209.48 762.58,-201.52 763.07,-212.1 769.56,-209.48\"/>\n",
"<text text-anchor=\"middle\" x=\"782.24\" y=\"-309.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->20 -->\n",
"<g id=\"edge41\" class=\"edge\">\n",
"<title>18->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M672.35,-112.95C694.14,-148.8 751.21,-227.94 784.56,-269.14\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"782.04,-271.59 791.1,-277.09 787.45,-267.15 782.04,-271.59\"/>\n",
"<text text-anchor=\"middle\" x=\"713.96\" y=\"-179.85\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->19 -->\n",
"<g id=\"edge42\" class=\"edge\">\n",
"<title>18->19</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M669.6,-113.05C692.45,-167.62 773.96,-328.19 810.95,-392.26\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"807.99,-394.13 816.07,-400.99 814.03,-390.59 807.99,-394.13\"/>\n",
"<text text-anchor=\"middle\" x=\"725.77\" y=\"-241.45\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->18 -->\n",
"<g id=\"edge43\" class=\"edge\">\n",
"<title>18->18</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M691.05,-101.34C700.84,-101.49 708.81,-99.24 708.81,-94.6 708.81,-91.77 705.85,-89.83 701.32,-88.78\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"701.33,-85.27 691.05,-87.85 700.7,-92.24 701.33,-85.27\"/>\n",
"<text text-anchor=\"middle\" x=\"723.31\" y=\"-90.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->17 -->\n",
"<g id=\"edge44\" class=\"edge\">\n",
"<title>18->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M641.06,-76.54C625.92,-67.55 605.72,-57.08 588.39,-49.16\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"589.66,-45.89 579.1,-45.04 586.82,-52.29 589.66,-45.89\"/>\n",
"<text text-anchor=\"middle\" x=\"600.22\" y=\"-51.65\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->16 -->\n",
"<g id=\"edge45\" class=\"edge\">\n",
"<title>18->16</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M636.79,-81C594.52,-64.37 512.68,-37.64 465,-24.76\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.65,-21.31 455.08,-22.15 463.87,-28.08 465.65,-21.31\"/>\n",
"<text text-anchor=\"middle\" x=\"536.4\" y=\"-56.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->15 -->\n",
"<g id=\"edge46\" class=\"edge\">\n",
"<title>18->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M636.78,-86.46C573.24,-72.6 414.3,-47.26 341.55,-39.21\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.77,-35.72 331.46,-38.16 341.04,-42.68 341.77,-35.72\"/>\n",
"<text text-anchor=\"middle\" x=\"474.66\" y=\"-51.64\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->14 -->\n",
"<g id=\"edge47\" class=\"edge\">\n",
"<title>18->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M636.77,-91.24C556.87,-86.89 321.73,-86.72 229.78,-90.73\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.37,-87.25 219.55,-91.23 229.71,-94.24 229.37,-87.25\"/>\n",
"<text text-anchor=\"middle\" x=\"418.78\" y=\"-79.79\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->13 -->\n",
"<g id=\"edge48\" class=\"edge\">\n",
"<title>18->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M636.69,-95.77C544.94,-105.54 246.11,-152.69 140.77,-173.8\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"140,-170.38 130.9,-175.81 141.4,-177.24 140,-170.38\"/>\n",
"<text text-anchor=\"middle\" x=\"374.23\" y=\"-123.58\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->12 -->\n",
"<g id=\"edge49\" class=\"edge\">\n",
"<title>18->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M636.78,-100.3C538.05,-127.22 196.89,-237.88 83.55,-279.5\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"82.08,-276.31 73.92,-283.07 84.52,-282.87 82.08,-276.31\"/>\n",
"<text text-anchor=\"middle\" x=\"345.67\" y=\"-178.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->11 -->\n",
"<g id=\"edge50\" class=\"edge\">\n",
"<title>18->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M636.76,-105.16C535.37,-151.22 178.15,-333.04 63.03,-396.9\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"61.24,-393.89 54.21,-401.82 64.65,-400 61.24,-393.89\"/>\n",
"<text text-anchor=\"middle\" x=\"335.39\" y=\"-239.83\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->10 -->\n",
"<g id=\"edge51\" class=\"edge\">\n",
"<title>18->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M636.77,-110.67C535.39,-178.07 178.82,-437.03 75.1,-518.45\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"72.63,-515.94 66.96,-524.89 76.98,-521.43 72.63,-515.94\"/>\n",
"<text text-anchor=\"middle\" x=\"341.43\" y=\"-303.36\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->9 -->\n",
"<g id=\"edge52\" class=\"edge\">\n",
"<title>18->9</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M641.96,-112.62C552.32,-194.74 214.08,-532.88 125.1,-629.17\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"122.37,-626.97 118.2,-636.71 127.53,-631.7 122.37,-626.97\"/>\n",
"<text text-anchor=\"middle\" x=\"369.03\" y=\"-374.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->8 -->\n",
"<g id=\"edge53\" class=\"edge\">\n",
"<title>18->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M647.4,-112.89C574.32,-204.06 280.22,-608.69 208.21,-716.89\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"205.25,-715.03 202.68,-725.3 211.1,-718.88 205.25,-715.03\"/>\n",
"<text text-anchor=\"middle\" x=\"413.31\" y=\"-403.69\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->7 -->\n",
"<g id=\"edge54\" class=\"edge\">\n",
"<title>18->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M651.87,-112.76C596.54,-208.84 367.77,-657.56 314.93,-773.26\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"311.68,-771.95 310.76,-782.51 318.06,-774.82 311.68,-771.95\"/>\n",
"<text text-anchor=\"middle\" x=\"468.9\" y=\"-431.81\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->6 -->\n",
"<g id=\"edge55\" class=\"edge\">\n",
"<title>18->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M655.39,-112.9C617.55,-210.86 467.25,-672.98 434.27,-792.09\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"430.84,-791.35 431.61,-801.92 437.6,-793.18 430.84,-791.35\"/>\n",
"<text text-anchor=\"middle\" x=\"530.33\" y=\"-456.3\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->5 -->\n",
"<g id=\"edge56\" class=\"edge\">\n",
"<title>18->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M658.55,-112.75C637.61,-208.58 566.64,-655.66 553.58,-772.51\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"550.1,-772.16 552.53,-782.47 557.06,-772.89 550.1,-772.16\"/>\n",
"<text text-anchor=\"middle\" x=\"591.57\" y=\"-446.43\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->4 -->\n",
"<g id=\"edge57\" class=\"edge\">\n",
"<title>18->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M661.39,-112.63C655.81,-202.77 655.63,-604.52 660.84,-715.22\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"657.35,-715.49 661.38,-725.29 664.34,-715.12 657.35,-715.49\"/>\n",
"<text text-anchor=\"middle\" x=\"646.62\" y=\"-417.73\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->3 -->\n",
"<g id=\"edge58\" class=\"edge\">\n",
"<title>18->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M664.08,-112.69C671.56,-194.19 724.14,-527.51 744.88,-626.79\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"741.48,-627.65 747.02,-636.68 748.33,-626.17 741.48,-627.65\"/>\n",
"<text text-anchor=\"middle\" x=\"689.98\" y=\"-373.54\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->2 -->\n",
"<g id=\"edge59\" class=\"edge\">\n",
"<title>18->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M666.76,-112.77C684.11,-182.37 765.01,-432.09 796.91,-515.6\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"793.66,-516.89 800.56,-524.94 800.18,-514.35 793.66,-516.89\"/>\n",
"<text text-anchor=\"middle\" x=\"717.34\" y=\"-317.99\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->1 -->\n",
"<g id=\"edge60\" class=\"edge\">\n",
"<title>18->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M674.81,-112.9C685.81,-126.1 703.08,-143.94 718.58,-158.41\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"716.27,-161.04 726.01,-165.2 720.99,-155.87 716.27,-161.04\"/>\n",
"<text text-anchor=\"middle\" x=\"682.19\" y=\"-139.46\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->20 -->\n",
"<g id=\"edge61\" class=\"edge\">\n",
"<title>17->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M565.29,-55.79C604.77,-100.99 721.43,-218.11 778.07,-269.94\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"775.88,-272.68 785.65,-276.8 780.58,-267.49 775.88,-272.68\"/>\n",
"<text text-anchor=\"middle\" x=\"657.18\" y=\"-151.66\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->19 -->\n",
"<g id=\"edge62\" class=\"edge\">\n",
"<title>17->19</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M561.53,-55.76C600.57,-117.62 747.39,-320.16 805.45,-392.86\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"803.07,-395.48 812.07,-401.06 808.51,-391.08 803.07,-395.48\"/>\n",
"<text text-anchor=\"middle\" x=\"668.99\" y=\"-213.11\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->18 -->\n",
"<g id=\"edge63\" class=\"edge\">\n",
"<title>17->18</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M574.76,-55.69C589.91,-64.67 610.1,-75.14 627.44,-83.07\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.17,-86.34 636.73,-87.19 629,-79.94 626.17,-86.34\"/>\n",
"<text text-anchor=\"middle\" x=\"586.6\" y=\"-73.18\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->17 -->\n",
"<g id=\"edge64\" class=\"edge\">\n",
"<title>17->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M579.25,-44.37C589.03,-44.52 597.01,-42.27 597.01,-37.63 597.01,-34.8 594.05,-32.86 589.52,-31.81\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"589.52,-28.3 579.25,-30.89 588.89,-35.27 589.52,-28.3\"/>\n",
"<text text-anchor=\"middle\" x=\"611.51\" y=\"-33.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->16 -->\n",
"<g id=\"edge65\" class=\"edge\">\n",
"<title>17->16</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M524.79,-27.49C507.37,-23.57 484.54,-19.8 465.44,-17.63\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.57,-14.13 455.27,-16.6 464.87,-21.09 465.57,-14.13\"/>\n",
"<text text-anchor=\"middle\" x=\"509.61\" y=\"-26.36\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->15 -->\n",
"<g id=\"edge66\" class=\"edge\">\n",
"<title>17->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M524.53,-33.2C479.93,-30.31 391.72,-30.12 341.47,-32.64\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.14,-29.15 331.35,-33.21 341.54,-36.14 341.14,-29.15\"/>\n",
"<text text-anchor=\"middle\" x=\"447.5\" y=\"-36.72\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->14 -->\n",
"<g id=\"edge67\" class=\"edge\">\n",
"<title>17->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M524.98,-38.13C461.45,-44.34 302.55,-69.3 229.78,-84.26\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"228.75,-80.9 219.69,-86.39 230.2,-87.75 228.75,-80.9\"/>\n",
"<text text-anchor=\"middle\" x=\"362.88\" y=\"-49.99\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->13 -->\n",
"<g id=\"edge68\" class=\"edge\">\n",
"<title>17->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M524.92,-42.88C447.91,-63.39 227.73,-134.72 140.15,-167.29\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"138.76,-164.07 130.63,-170.87 141.23,-170.62 138.76,-164.07\"/>\n",
"<text text-anchor=\"middle\" x=\"318.04\" y=\"-93.88\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->12 -->\n",
"<g id=\"edge69\" class=\"edge\">\n",
"<title>17->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M524.8,-47.91C439.73,-86.16 178.85,-218.87 82.69,-272.54\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"80.91,-269.52 73.91,-277.48 84.34,-275.62 80.91,-269.52\"/>\n",
"<text text-anchor=\"middle\" x=\"289.25\" y=\"-149.03\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->11 -->\n",
"<g id=\"edge70\" class=\"edge\">\n",
"<title>17->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M524.86,-53.51C434.76,-113.08 146.51,-322.4 55.07,-394.53\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"52.74,-391.91 47.1,-400.88 57.1,-397.39 52.74,-391.91\"/>\n",
"<text text-anchor=\"middle\" x=\"275.46\" y=\"-212.82\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->10 -->\n",
"<g id=\"edge71\" class=\"edge\">\n",
"<title>17->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M529.87,-55.74C446.14,-132.21 150.1,-428.16 67.89,-517.43\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"65.19,-515.19 61.04,-524.94 70.36,-519.91 65.19,-515.19\"/>\n",
"<text text-anchor=\"middle\" x=\"284.38\" y=\"-290.38\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->9 -->\n",
"<g id=\"edge72\" class=\"edge\">\n",
"<title>17->9</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M535.5,-55.94C464.74,-144.06 188.71,-523.83 119.38,-628.21\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"116.42,-626.34 113.85,-636.62 122.27,-630.18 116.42,-626.34\"/>\n",
"<text text-anchor=\"middle\" x=\"312.94\" y=\"-345.88\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->8 -->\n",
"<g id=\"edge73\" class=\"edge\">\n",
"<title>17->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M540.06,-55.79C484.73,-151.87 255.97,-600.59 203.12,-716.29\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"199.87,-714.98 198.96,-725.54 206.25,-717.86 199.87,-714.98\"/>\n",
"<text text-anchor=\"middle\" x=\"357.09\" y=\"-389.84\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->7 -->\n",
"<g id=\"edge74\" class=\"edge\">\n",
"<title>17->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M543.7,-55.78C504.79,-156.77 344.65,-649.16 310.39,-772.48\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"306.96,-771.73 307.72,-782.3 313.72,-773.57 306.96,-771.73\"/>\n",
"<text text-anchor=\"middle\" x=\"441.54\" y=\"-417.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->6 -->\n",
"<g id=\"edge75\" class=\"edge\">\n",
"<title>17->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M546.84,-55.85C524.63,-158.72 444.02,-666.7 429.77,-792.11\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"426.29,-791.77 428.7,-802.08 433.25,-792.52 426.29,-791.77\"/>\n",
"<text text-anchor=\"middle\" x=\"502.81\" y=\"-427.78\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->5 -->\n",
"<g id=\"edge76\" class=\"edge\">\n",
"<title>17->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M549.76,-55.74C543.94,-156.49 543.76,-647.97 549.24,-772.03\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"545.75,-772.42 549.75,-782.24 552.74,-772.07 545.75,-772.42\"/>\n",
"<text text-anchor=\"middle\" x=\"535\" y=\"-417.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->4 -->\n",
"<g id=\"edge77\" class=\"edge\">\n",
"<title>17->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M552.54,-55.79C561.93,-151.66 632.55,-598.73 656.48,-715.55\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"653.09,-716.45 658.58,-725.51 659.94,-715 653.09,-716.45\"/>\n",
"<text text-anchor=\"middle\" x=\"590.01\" y=\"-389.47\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->3 -->\n",
"<g id=\"edge78\" class=\"edge\">\n",
"<title>17->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M555.36,-55.86C578.02,-143.48 700.53,-521.21 740.33,-627.12\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"737.18,-628.69 744.03,-636.78 743.72,-626.19 737.18,-628.69\"/>\n",
"<text text-anchor=\"middle\" x=\"633.35\" y=\"-345.29\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->2 -->\n",
"<g id=\"edge79\" class=\"edge\">\n",
"<title>17->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M558.23,-55.64C590.98,-131.47 740.13,-424.71 791.92,-515.85\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"789.02,-517.83 797.05,-524.75 795.09,-514.33 789.02,-517.83\"/>\n",
"<text text-anchor=\"middle\" x=\"660.58\" y=\"-289.54\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->1 -->\n",
"<g id=\"edge80\" class=\"edge\">\n",
"<title>17->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M569.83,-55.86C602.82,-83.52 673.74,-135.4 716.91,-163.79\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"715.12,-166.8 725.42,-169.31 718.93,-160.93 715.12,-166.8\"/>\n",
"<text text-anchor=\"middle\" x=\"628.87\" y=\"-113.63\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->20 -->\n",
"<g id=\"edge81\" class=\"edge\">\n",
"<title>16->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M447.44,-36.18C508.13,-85.61 694.76,-221.5 773.71,-274.1\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"772.07,-277.21 782.34,-279.8 775.93,-271.37 772.07,-277.21\"/>\n",
"<text text-anchor=\"middle\" x=\"596.08\" y=\"-143.94\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->19 -->\n",
"<g id=\"edge82\" class=\"edge\">\n",
"<title>16->19</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M442.26,-36.25C500.07,-100.99 717.76,-319.05 798.68,-393.77\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"796.78,-396.78 806.52,-400.95 801.51,-391.62 796.78,-396.78\"/>\n",
"<text text-anchor=\"middle\" x=\"605.97\" y=\"-203.81\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->18 -->\n",
"<g id=\"edge83\" class=\"edge\">\n",
"<title>16->18</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M455.09,-31.6C497.37,-48.22 579.2,-74.95 626.88,-87.84\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.24,-91.29 636.8,-90.45 628.02,-84.52 626.24,-91.29\"/>\n",
"<text text-anchor=\"middle\" x=\"555.49\" y=\"-63.52\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->17 -->\n",
"<g id=\"edge84\" class=\"edge\">\n",
"<title>16->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M455.29,-28.14C472.71,-32.06 495.53,-35.83 514.64,-38\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"514.51,-41.5 524.81,-39.03 515.21,-34.54 514.51,-41.5\"/>\n",
"<text text-anchor=\"middle\" x=\"499.46\" y=\"-36.87\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->16 -->\n",
"<g id=\"edge85\" class=\"edge\">\n",
"<title>16->16</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M455.31,-24.74C465.09,-24.89 473.07,-22.64 473.07,-18 473.07,-15.17 470.11,-13.23 465.58,-12.18\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.59,-8.67 455.31,-11.26 464.96,-15.64 465.59,-8.67\"/>\n",
"<text text-anchor=\"middle\" x=\"487.57\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->15 -->\n",
"<g id=\"edge86\" class=\"edge\">\n",
"<title>16->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M400.9,-16.6C383.37,-18.13 360.33,-21.57 341.13,-25.41\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"340.26,-22.02 331.2,-27.52 341.71,-28.87 340.26,-22.02\"/>\n",
"<text text-anchor=\"middle\" x=\"356.52\" y=\"-9.8\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->14 -->\n",
"<g id=\"edge87\" class=\"edge\">\n",
"<title>16->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M401.03,-22.16C358.7,-32.9 276.75,-59.27 229.1,-77.25\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"227.56,-74.09 219.48,-80.95 230.07,-80.62 227.56,-74.09\"/>\n",
"<text text-anchor=\"middle\" x=\"329.57\" y=\"-53.5\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->13 -->\n",
"<g id=\"edge88\" class=\"edge\">\n",
"<title>16->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M400.86,-27.55C342.28,-53.31 204.35,-123.34 139.04,-160.21\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"137.27,-157.19 130.32,-165.18 140.74,-163.27 137.27,-157.19\"/>\n",
"<text text-anchor=\"middle\" x=\"284.45\" y=\"-97.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->12 -->\n",
"<g id=\"edge89\" class=\"edge\">\n",
"<title>16->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M400.95,-33.3C330.81,-79.11 144.29,-214.52 73.94,-270.56\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"71.55,-268 65.95,-276.99 75.94,-273.45 71.55,-268\"/>\n",
"<text text-anchor=\"middle\" x=\"222.95\" y=\"-140.73\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->11 -->\n",
"<g id=\"edge90\" class=\"edge\">\n",
"<title>16->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M405.38,-36.19C334.3,-100.65 117.18,-317.67 48.2,-393.07\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"45.38,-390.97 41.26,-400.74 50.57,-395.67 45.38,-390.97\"/>\n",
"<text text-anchor=\"middle\" x=\"212.29\" y=\"-218.43\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->10 -->\n",
"<g id=\"edge91\" class=\"edge\">\n",
"<title>16->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M411.32,-36.3C347.87,-114.83 123.86,-422.99 62.29,-516.2\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"59.22,-514.49 56.69,-524.78 65.09,-518.31 59.22,-514.49\"/>\n",
"<text text-anchor=\"middle\" x=\"222.31\" y=\"-280.05\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->9 -->\n",
"<g id=\"edge92\" class=\"edge\">\n",
"<title>16->9</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M415.93,-36.26C364.06,-125.93 163.46,-519.37 114.35,-627.34\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"111.09,-626.04 110.2,-636.6 117.48,-628.9 111.09,-626.04\"/>\n",
"<text text-anchor=\"middle\" x=\"250.64\" y=\"-335.6\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->8 -->\n",
"<g id=\"edge93\" class=\"edge\">\n",
"<title>16->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M419.64,-36.3C381.81,-134.27 231.51,-596.39 198.52,-715.49\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"195.1,-714.75 195.86,-725.32 201.85,-716.58 195.1,-714.75\"/>\n",
"<text text-anchor=\"middle\" x=\"294.58\" y=\"-379.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->7 -->\n",
"<g id=\"edge94\" class=\"edge\">\n",
"<title>16->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M422.91,-36.22C400.69,-139.09 320.09,-647.07 305.84,-772.48\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"302.35,-772.14 304.76,-782.45 309.31,-772.89 302.35,-772.14\"/>\n",
"<text text-anchor=\"middle\" x=\"349.87\" y=\"-408.15\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->6 -->\n",
"<g id=\"edge95\" class=\"edge\">\n",
"<title>16->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M425.86,-36.26C419.98,-140.84 419.81,-663.54 425.36,-791.73\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"421.88,-792.09 425.86,-801.91 428.87,-791.75 421.88,-792.09\"/>\n",
"<text text-anchor=\"middle\" x=\"411.11\" y=\"-417.79\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->5 -->\n",
"<g id=\"edge96\" class=\"edge\">\n",
"<title>16->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M428.71,-36.16C439.03,-138.87 519.31,-646.83 544.78,-772.4\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"541.39,-773.31 546.86,-782.39 548.25,-771.88 541.39,-773.31\"/>\n",
"<text text-anchor=\"middle\" x=\"501.25\" y=\"-408.08\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->4 -->\n",
"<g id=\"edge97\" class=\"edge\">\n",
"<title>16->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M431.62,-36.28C457.39,-134.27 607.6,-597.18 651.81,-715.8\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"648.59,-717.17 655.4,-725.28 655.13,-714.69 648.59,-717.17\"/>\n",
"<text text-anchor=\"middle\" x=\"527.22\" y=\"-379.84\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->3 -->\n",
"<g id=\"edge98\" class=\"edge\">\n",
"<title>16->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M434.63,-36.11C473.95,-125.64 674.89,-520.47 735.42,-627.81\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"732.51,-629.78 740.51,-636.73 738.59,-626.31 732.51,-629.78\"/>\n",
"<text text-anchor=\"middle\" x=\"570.52\" y=\"-335.76\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->2 -->\n",
"<g id=\"edge99\" class=\"edge\">\n",
"<title>16->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M438.04,-36.09C488.37,-114.44 713.12,-424.17 786.42,-516.77\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"783.85,-519.16 792.83,-524.79 789.32,-514.79 783.85,-519.16\"/>\n",
"<text text-anchor=\"middle\" x=\"597.73\" y=\"-280.23\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->1 -->\n",
"<g id=\"edge100\" class=\"edge\">\n",
"<title>16->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M454.75,-36.12C512.7,-69.59 650.03,-139.69 715.99,-169.6\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"714.63,-172.82 725.18,-173.71 717.48,-166.43 714.63,-172.82\"/>\n",
"<text text-anchor=\"middle\" x=\"570.87\" y=\"-106.66\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->20 -->\n",
"<g id=\"edge101\" class=\"edge\">\n",
"<title>15->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M331.24,-55.19C416.12,-103.43 676.74,-236.33 773.17,-280.69\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"771.96,-283.98 782.51,-284.94 774.85,-277.61 771.96,-283.98\"/>\n",
"<text text-anchor=\"middle\" x=\"537.7\" y=\"-171.74\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->19 -->\n",
"<g id=\"edge102\" class=\"edge\">\n",
"<title>15->19</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M323.99,-55.63C402.56,-118.81 691.88,-329.27 793.56,-397.57\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"791.7,-400.53 801.96,-403.17 795.58,-394.71 791.7,-400.53\"/>\n",
"<text text-anchor=\"middle\" x=\"544.28\" y=\"-215.4\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->18 -->\n",
"<g id=\"edge103\" class=\"edge\">\n",
"<title>15->18</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M331.16,-45.77C394.7,-59.63 553.65,-84.97 626.4,-93.01\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.18,-96.51 636.49,-94.07 626.91,-89.55 626.18,-96.51\"/>\n",
"<text text-anchor=\"middle\" x=\"464.28\" y=\"-73.19\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->17 -->\n",
"<g id=\"edge104\" class=\"edge\">\n",
"<title>15->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M331.61,-42.06C376.21,-44.95 464.42,-45.14 514.67,-42.62\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"515,-46.11 524.79,-42.05 514.61,-39.12 515,-46.11\"/>\n",
"<text text-anchor=\"middle\" x=\"408.64\" y=\"-46.14\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->16 -->\n",
"<g id=\"edge105\" class=\"edge\">\n",
"<title>15->16</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M331.3,-39.03C348.84,-37.5 371.88,-34.06 391.08,-30.22\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"391.95,-33.61 401,-28.11 390.49,-26.76 391.95,-33.61\"/>\n",
"<text text-anchor=\"middle\" x=\"346.69\" y=\"-23.42\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->15 -->\n",
"<g id=\"edge106\" class=\"edge\">\n",
"<title>15->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M331.37,-44.37C341.16,-44.52 349.13,-42.27 349.13,-37.63 349.13,-34.8 346.17,-32.86 341.64,-31.81\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.65,-28.3 331.37,-30.89 341.02,-35.27 341.65,-28.3\"/>\n",
"<text text-anchor=\"middle\" x=\"363.63\" y=\"-33.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->14 -->\n",
"<g id=\"edge107\" class=\"edge\">\n",
"<title>15->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M277.01,-45.05C261.08,-51.87 240.89,-61.93 224.2,-71.28\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"222.27,-68.36 215.35,-76.38 225.76,-74.43 222.27,-68.36\"/>\n",
"<text text-anchor=\"middle\" x=\"236.11\" y=\"-61.97\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->13 -->\n",
"<g id=\"edge108\" class=\"edge\">\n",
"<title>15->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M276.83,-51.76C237.97,-76.59 167.38,-127.77 129.38,-158.55\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"126.84,-156.1 121.34,-165.16 131.28,-161.51 126.84,-156.1\"/>\n",
"<text text-anchor=\"middle\" x=\"217.6\" y=\"-108.95\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->12 -->\n",
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"<title>15->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M280.29,-55.94C229.88,-101.07 113.63,-217.25 66.7,-269.33\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"63.81,-267.31 59.79,-277.12 69.04,-271.96 63.81,-267.31\"/>\n",
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"</g>\n",
"<!-- 15->11 -->\n",
"<g id=\"edge110\" class=\"edge\">\n",
"<title>15->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M287.01,-55.71C236.63,-117.25 90.33,-318.43 42.32,-391.93\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"39.18,-390.33 36.71,-400.64 45.07,-394.12 39.18,-390.33\"/>\n",
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"</g>\n",
"<!-- 15->10 -->\n",
"<g id=\"edge111\" class=\"edge\">\n",
"<title>15->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M291.69,-55.85C247.17,-131.95 98.09,-424.26 57.01,-515.6\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"53.69,-514.44 52.85,-525 60.1,-517.27 53.69,-514.44\"/>\n",
"<text text-anchor=\"middle\" x=\"159.85\" y=\"-289.53\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->9 -->\n",
"<g id=\"edge112\" class=\"edge\">\n",
"<title>15->9</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M295.62,-55.66C261.54,-142.82 138.64,-520.58 109.54,-626.89\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"106.15,-626.02 106.95,-636.59 112.91,-627.83 106.15,-626.02\"/>\n",
"<text text-anchor=\"middle\" x=\"188.08\" y=\"-345.07\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->8 -->\n",
"<g id=\"edge113\" class=\"edge\">\n",
"<title>15->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M298.87,-55.78C277.93,-151.61 206.96,-598.7 193.9,-715.54\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"190.42,-715.19 192.85,-725.5 197.38,-715.92 190.42,-715.19\"/>\n",
"<text text-anchor=\"middle\" x=\"231.88\" y=\"-389.46\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->7 -->\n",
"<g id=\"edge114\" class=\"edge\">\n",
"<title>15->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M301.88,-55.74C296.06,-156.49 295.89,-647.97 301.36,-772.03\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"297.88,-772.42 301.87,-782.24 304.87,-772.07 297.88,-772.42\"/>\n",
"<text text-anchor=\"middle\" x=\"287.12\" y=\"-417.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->6 -->\n",
"<g id=\"edge115\" class=\"edge\">\n",
"<title>15->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M304.77,-55.79C315.1,-158.5 395.37,-666.46 420.84,-792.03\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"417.46,-792.94 422.93,-802.02 424.31,-791.51 417.46,-792.94\"/>\n",
"<text text-anchor=\"middle\" x=\"348.31\" y=\"-427.71\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->5 -->\n",
"<g id=\"edge116\" class=\"edge\">\n",
"<title>15->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M307.68,-55.68C334.28,-156.51 494.34,-649.74 540.05,-772.71\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"536.82,-774.07 543.63,-782.18 543.37,-771.59 536.82,-774.07\"/>\n",
"<text text-anchor=\"middle\" x=\"438.36\" y=\"-402.99\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->4 -->\n",
"<g id=\"edge117\" class=\"edge\">\n",
"<title>15->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M310.82,-55.76C353.32,-151.87 582.11,-601.33 646.86,-716.58\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"643.95,-718.53 651.93,-725.49 650.03,-715.07 643.95,-718.53\"/>\n",
"<text text-anchor=\"middle\" x=\"464.34\" y=\"-374.97\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->3 -->\n",
"<g id=\"edge118\" class=\"edge\">\n",
"<title>15->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M314.42,-55.87C371.54,-144 648.17,-525.09 729.87,-628.74\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"727.32,-631.16 736.29,-636.81 732.8,-626.81 727.32,-631.16\"/>\n",
"<text text-anchor=\"middle\" x=\"507.64\" y=\"-346.1\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->2 -->\n",
"<g id=\"edge119\" class=\"edge\">\n",
"<title>15->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M318.54,-55.7C387.59,-132.26 684.13,-429.12 779.73,-517.87\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"777.59,-520.65 787.31,-524.85 782.33,-515.5 777.59,-520.65\"/>\n",
"<text text-anchor=\"middle\" x=\"534.64\" y=\"-290.59\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 15->1 -->\n",
"<g id=\"edge120\" class=\"edge\">\n",
"<title>15->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M331.44,-50.19C408.57,-79.67 628.05,-151.11 715.72,-175.43\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"714.93,-178.84 725.5,-178.09 716.77,-172.08 714.93,-178.84\"/>\n",
"<text text-anchor=\"middle\" x=\"509.08\" y=\"-116.61\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->20 -->\n",
"<g id=\"edge121\" class=\"edge\">\n",
"<title>14->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.51,-106.62C318.41,-143.84 659.06,-254.65 772.47,-286.65\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"771.84,-290.11 782.41,-289.41 773.71,-283.36 771.84,-290.11\"/>\n",
"<text text-anchor=\"middle\" x=\"481.49\" y=\"-200.44\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->19 -->\n",
"<g id=\"edge122\" class=\"edge\">\n",
"<title>14->19</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.56,-111.85C321.03,-169.04 677.06,-350.55 792.63,-404.17\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"791.25,-407.38 801.8,-408.38 794.17,-401.02 791.25,-407.38\"/>\n",
"<text text-anchor=\"middle\" x=\"491.59\" y=\"-261.81\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->18 -->\n",
"<g id=\"edge123\" class=\"edge\">\n",
"<title>14->18</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.37,-97.95C299.27,-102.3 534.41,-102.47 626.36,-98.46\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.77,-101.95 636.59,-97.96 626.43,-94.95 626.77,-101.95\"/>\n",
"<text text-anchor=\"middle\" x=\"408.36\" y=\"-102.01\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->17 -->\n",
"<g id=\"edge124\" class=\"edge\">\n",
"<title>14->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.36,-94.1C282.89,-87.89 441.78,-62.93 514.55,-47.97\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"515.58,-51.33 524.64,-45.84 514.13,-44.48 515.58,-51.33\"/>\n",
"<text text-anchor=\"middle\" x=\"352.46\" y=\"-74.84\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->16 -->\n",
"<g id=\"edge125\" class=\"edge\">\n",
"<title>14->16</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.36,-90.44C261.69,-79.7 343.64,-53.33 391.3,-35.35\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"392.84,-38.51 400.92,-31.65 390.33,-31.97 392.84,-38.51\"/>\n",
"<text text-anchor=\"middle\" x=\"290.83\" y=\"-66.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->15 -->\n",
"<g id=\"edge126\" class=\"edge\">\n",
"<title>14->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.45,-87.17C235.38,-80.36 255.57,-70.29 272.26,-60.94\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"274.19,-63.87 281.11,-55.85 270.7,-57.8 274.19,-63.87\"/>\n",
"<text text-anchor=\"middle\" x=\"260.35\" y=\"-77.86\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->14 -->\n",
"<g id=\"edge127\" class=\"edge\">\n",
"<title>14->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.57,-101.34C229.35,-101.49 237.33,-99.24 237.33,-94.6 237.33,-91.77 234.37,-89.83 229.84,-88.78\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.84,-85.27 219.57,-87.85 229.21,-92.24 229.84,-85.27\"/>\n",
"<text text-anchor=\"middle\" x=\"251.83\" y=\"-90.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->13 -->\n",
"<g id=\"edge128\" class=\"edge\">\n",
"<title>14->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M165.73,-112.78C151.27,-125.7 133.87,-143.14 121.1,-157.48\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"118.42,-155.23 114.54,-165.09 123.72,-159.81 118.42,-155.23\"/>\n",
"<text text-anchor=\"middle\" x=\"128.92\" y=\"-138.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->12 -->\n",
"<g id=\"edge129\" class=\"edge\">\n",
"<title>14->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M173.8,-112.77C144.07,-148.31 86.87,-226.88 60.42,-268.37\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"57.29,-266.77 54.97,-277.11 63.23,-270.47 57.29,-266.77\"/>\n",
"<text text-anchor=\"middle\" x=\"131.61\" y=\"-194.37\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->11 -->\n",
"<g id=\"edge130\" class=\"edge\">\n",
"<title>14->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M179.16,-112.84C146.82,-166.94 65.36,-326.5 36.8,-391.29\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"33.43,-390.28 32.69,-400.84 39.86,-393.04 33.43,-390.28\"/>\n",
"<text text-anchor=\"middle\" x=\"93.48\" y=\"-255.87\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->10 -->\n",
"<g id=\"edge131\" class=\"edge\">\n",
"<title>14->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M183.37,-112.67C155.68,-181.97 74.64,-430.87 52.13,-515\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"48.69,-514.31 49.57,-524.87 55.47,-516.07 48.69,-514.31\"/>\n",
"<text text-anchor=\"middle\" x=\"103.25\" y=\"-317.64\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->9 -->\n",
"<g id=\"edge132\" class=\"edge\">\n",
"<title>14->9</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M186.78,-112.83C168.47,-194.43 115.69,-526.55 104.91,-626.33\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"101.4,-626.25 103.88,-636.55 108.37,-626.95 101.4,-626.25\"/>\n",
"<text text-anchor=\"middle\" x=\"131.34\" y=\"-373.38\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->8 -->\n",
"<g id=\"edge133\" class=\"edge\">\n",
"<title>14->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M189.91,-112.63C184.32,-202.77 184.14,-604.52 189.35,-715.22\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"185.86,-715.49 189.89,-725.29 192.85,-715.12 185.86,-715.49\"/>\n",
"<text text-anchor=\"middle\" x=\"175.13\" y=\"-417.73\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->7 -->\n",
"<g id=\"edge134\" class=\"edge\">\n",
"<title>14->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M192.86,-112.76C202.25,-208.63 272.87,-655.7 296.8,-772.52\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"293.41,-773.42 298.9,-782.48 300.26,-771.97 293.41,-773.42\"/>\n",
"<text text-anchor=\"middle\" x=\"230.33\" y=\"-446.44\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->6 -->\n",
"<g id=\"edge135\" class=\"edge\">\n",
"<title>14->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M195.88,-112.88C221.64,-210.86 371.85,-673.77 416.07,-792.4\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"412.84,-793.77 419.66,-801.88 419.39,-791.29 412.84,-793.77\"/>\n",
"<text text-anchor=\"middle\" x=\"291.47\" y=\"-456.44\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->5 -->\n",
"<g id=\"edge136\" class=\"edge\">\n",
"<title>14->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M199.02,-112.73C241.51,-208.84 470.31,-658.3 535.06,-773.55\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"532.14,-775.49 540.12,-782.46 538.23,-772.04 532.14,-775.49\"/>\n",
"<text text-anchor=\"middle\" x=\"352.54\" y=\"-446.94\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 14->4 -->\n",
"<g id=\"edge137\" class=\"edge\">\n",
"<title>14->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M202.57,-112.66C261.55,-203.5 556.31,-609.54 641.07,-717.27\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"638.52,-719.7 647.48,-725.35 644.01,-715.34 638.52,-719.7\"/>\n",
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"</g>\n",
"<!-- 14->3 -->\n",
"<g id=\"edge138\" class=\"edge\">\n",
"<title>14->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M207.03,-112.82C281.91,-195.42 620.18,-533.99 722.91,-629.61\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"720.74,-632.36 730.46,-636.58 725.49,-627.22 720.74,-632.36\"/>\n",
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"</g>\n",
"<!-- 14->2 -->\n",
"<g id=\"edge139\" class=\"edge\">\n",
"<title>14->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M212.7,-112.75C302.05,-184.08 659.43,-443.96 774,-521.26\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"772.2,-524.27 782.45,-526.93 776.1,-518.45 772.2,-524.27\"/>\n",
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"</g>\n",
"<!-- 14->1 -->\n",
"<g id=\"edge140\" class=\"edge\">\n",
"<title>14->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.62,-102.11C311.52,-121.38 609.89,-168.78 715.27,-181.03\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"715.1,-184.53 725.43,-182.16 715.88,-177.57 715.1,-184.53\"/>\n",
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"<!-- 13->20 -->\n",
"<g id=\"edge141\" class=\"edge\">\n",
"<title>13->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M130.83,-190.51C240.01,-212.95 646.19,-277.42 772.07,-292.47\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"772.02,-295.98 782.35,-293.66 772.82,-289.03 772.02,-295.98\"/>\n",
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"<!-- 13->19 -->\n",
"<g id=\"edge142\" class=\"edge\">\n",
"<title>13->19</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M130.73,-195.1C242.1,-236.69 664.26,-373.97 792.19,-410.37\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"791.35,-413.77 801.92,-413.11 793.24,-407.03 791.35,-413.77\"/>\n",
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"<!-- 13->18 -->\n",
"<g id=\"edge143\" class=\"edge\">\n",
"<title>13->18</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M130.72,-182.16C222.48,-172.39 521.3,-125.23 626.65,-104.13\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"627.42,-107.54 636.51,-102.11 626.01,-100.69 627.42,-107.54\"/>\n",
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"<!-- 13->17 -->\n",
"<g id=\"edge144\" class=\"edge\">\n",
"<title>13->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M130.69,-178.07C207.7,-157.57 427.88,-86.23 515.45,-53.67\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"516.85,-56.89 524.97,-50.08 514.38,-50.34 516.85,-56.89\"/>\n",
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"<!-- 13->16 -->\n",
"<g id=\"edge145\" class=\"edge\">\n",
"<title>13->16</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"394.4,-44.14 401.35,-36.14 390.93,-38.06 394.4,-44.14\"/>\n",
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"<!-- 13->15 -->\n",
"<g id=\"edge146\" class=\"edge\">\n",
"<title>13->15</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"280.89,-64.85 286.39,-55.8 276.45,-59.45 280.89,-64.85\"/>\n",
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"</g>\n",
"<!-- 13->14 -->\n",
"<g id=\"edge147\" class=\"edge\">\n",
"<title>13->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M130.2,-165.15C144.65,-152.22 162.06,-134.79 174.82,-120.44\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"177.5,-122.69 181.38,-112.83 172.2,-118.12 177.5,-122.69\"/>\n",
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"<!-- 13->13 -->\n",
"<g id=\"edge148\" class=\"edge\">\n",
"<title>13->13</title>\n",
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"<title>13->12</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"50.99,-266.29 50.43,-276.87 57.46,-268.95 50.99,-266.29\"/>\n",
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"<title>13->11</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"27.96,-390.4 28.99,-400.94 34.76,-392.06 27.96,-390.4\"/>\n",
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"</g>\n",
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"<title>13->10</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"43.76,-514.45 46.34,-524.73 50.73,-515.08 43.76,-514.45\"/>\n",
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"</g>\n",
"<!-- 13->9 -->\n",
"<g id=\"edge152\" class=\"edge\">\n",
"<title>13->9</title>\n",
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"<title>13->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M103.86,-201.42C111.34,-282.92 163.93,-616.24 184.66,-715.52\"/>\n",
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"</g>\n",
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"<title>13->7</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"288.77,-774.38 295.61,-782.47 295.31,-771.88 288.77,-774.38\"/>\n",
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"</g>\n",
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"<title>13->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M110.16,-201.44C149.47,-290.96 350.42,-685.8 410.95,-793.14\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"408.04,-795.11 416.03,-802.06 414.12,-791.64 408.04,-795.11\"/>\n",
"<text text-anchor=\"middle\" x=\"246.05\" y=\"-501.09\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<g id=\"edge156\" class=\"edge\">\n",
"<title>13->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M113.88,-201.57C171.01,-289.69 447.63,-670.79 529.33,-774.44\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"526.79,-776.86 535.75,-782.51 532.27,-772.5 526.79,-776.86\"/>\n",
"<text text-anchor=\"middle\" x=\"307.11\" y=\"-491.8\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 13->4 -->\n",
"<g id=\"edge157\" class=\"edge\">\n",
"<title>13->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M118.3,-201.55C193.18,-284.15 531.45,-622.72 634.19,-718.34\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"632.01,-721.09 641.73,-725.31 636.76,-715.95 632.01,-721.09\"/>\n",
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"</g>\n",
"<!-- 13->3 -->\n",
"<g id=\"edge158\" class=\"edge\">\n",
"<title>13->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M124,-201.43C216.7,-275.29 598.23,-552.72 717.07,-633\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"715.23,-635.99 725.49,-638.66 719.13,-630.18 715.23,-635.99\"/>\n",
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"</g>\n",
"<!-- 13->2 -->\n",
"<g id=\"edge159\" class=\"edge\">\n",
"<title>13->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M130.84,-200.43C240.4,-261.96 648.74,-470.12 773.25,-528.06\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"771.85,-531.27 782.4,-532.28 774.78,-524.91 771.85,-531.27\"/>\n",
"<text text-anchor=\"middle\" x=\"437.55\" y=\"-368.05\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 13->1 -->\n",
"<g id=\"edge160\" class=\"edge\">\n",
"<title>13->1</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"715.53,-190.18 725.37,-186.25 715.23,-183.19 715.53,-190.18\"/>\n",
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"</g>\n",
"<!-- 12->20 -->\n",
"<g id=\"edge161\" class=\"edge\">\n",
"<title>12->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.78,-297.84C189.17,-303.09 639.02,-303.23 772.18,-298.26\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"772.44,-301.75 782.29,-297.85 772.15,-294.76 772.44,-301.75\"/>\n",
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"</g>\n",
"<!-- 12->19 -->\n",
"<g id=\"edge162\" class=\"edge\">\n",
"<title>12->19</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"791.68,-419.78 802.02,-417.47 792.49,-412.83 791.68,-419.78\"/>\n",
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"</g>\n",
"<!-- 12->18 -->\n",
"<g id=\"edge163\" class=\"edge\">\n",
"<title>12->18</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.66,-289.43C172.39,-262.51 513.55,-151.85 626.89,-110.23\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"628.36,-113.42 636.52,-106.66 625.93,-106.86 628.36,-113.42\"/>\n",
"<text text-anchor=\"middle\" x=\"335.78\" y=\"-203.63\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 12->17 -->\n",
"<g id=\"edge164\" class=\"edge\">\n",
"<title>12->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.83,-284.85C158.91,-246.61 419.79,-113.9 515.95,-60.23\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"517.73,-63.24 524.73,-55.29 514.3,-57.14 517.73,-63.24\"/>\n",
"<text text-anchor=\"middle\" x=\"280.39\" y=\"-176.34\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 12->16 -->\n",
"<g id=\"edge165\" class=\"edge\">\n",
"<title>12->16</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.75,-279.83C143.89,-234.02 330.41,-98.61 400.76,-42.57\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"403.15,-45.14 408.75,-36.14 398.76,-39.68 403.15,-45.14\"/>\n",
"<text text-anchor=\"middle\" x=\"222.75\" y=\"-165\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 12->15 -->\n",
"<g id=\"edge166\" class=\"edge\">\n",
"<title>12->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M70.47,-276.83C120.88,-231.7 237.14,-115.51 284.06,-63.43\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"286.95,-65.45 290.98,-55.65 281.72,-60.8 286.95,-65.45\"/>\n",
"<text text-anchor=\"middle\" x=\"191.76\" y=\"-173.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 12->14 -->\n",
"<g id=\"edge167\" class=\"edge\">\n",
"<title>12->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M65.16,-276.96C94.89,-241.42 152.08,-162.85 178.54,-121.37\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"181.67,-122.96 183.99,-112.62 175.73,-119.26 181.67,-122.96\"/>\n",
"<text text-anchor=\"middle\" x=\"107.35\" y=\"-202.96\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 12->13 -->\n",
"<g id=\"edge168\" class=\"edge\">\n",
"<title>12->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M61.86,-277.06C72.88,-259.2 87.03,-231.71 95.9,-211.09\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"99.24,-212.17 99.8,-201.59 92.76,-209.51 99.24,-212.17\"/>\n",
"<text text-anchor=\"middle\" x=\"93.38\" y=\"-232.87\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 12->12 -->\n",
"<g id=\"edge169\" class=\"edge\">\n",
"<title>12->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.87,-301.88C83.65,-302.02 91.63,-299.77 91.63,-295.13 91.63,-292.3 88.67,-290.37 84.14,-289.31\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"84.14,-285.8 73.87,-288.39 83.52,-292.77 84.14,-285.8\"/>\n",
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"</g>\n",
"<!-- 12->11 -->\n",
"<g id=\"edge170\" class=\"edge\">\n",
"<title>12->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M38.7,-313.17C33.4,-333.52 27.88,-366.87 25.7,-390.71\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"22.2,-390.68 24.95,-400.91 29.18,-391.2 22.2,-390.68\"/>\n",
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"</g>\n",
"<!-- 12->10 -->\n",
"<g id=\"edge171\" class=\"edge\">\n",
"<title>12->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M42.89,-313.36C39.14,-356.16 38.89,-463.32 42.14,-514.68\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"38.66,-515.04 42.89,-524.75 45.64,-514.52 38.66,-515.04\"/>\n",
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"</g>\n",
"<!-- 12->9 -->\n",
"<g id=\"edge172\" class=\"edge\">\n",
"<title>12->9</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M46.33,-313.21C50.96,-371.61 79.82,-555.48 95.17,-626.46\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"91.82,-627.48 97.43,-636.47 98.65,-625.95 91.82,-627.48\"/>\n",
"<text text-anchor=\"middle\" x=\"56.25\" y=\"-473.64\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 12->8 -->\n",
"<g id=\"edge173\" class=\"edge\">\n",
"<title>12->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M49.58,-313.31C66.93,-382.9 147.82,-632.63 179.73,-716.14\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"176.48,-717.43 183.37,-725.47 183,-714.88 176.48,-717.43\"/>\n",
"<text text-anchor=\"middle\" x=\"100.15\" y=\"-518.52\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 12->7 -->\n",
"<g id=\"edge174\" class=\"edge\">\n",
"<title>12->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M52.85,-313.14C85.6,-388.97 234.75,-682.21 286.54,-773.35\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"283.64,-775.33 291.67,-782.25 289.71,-771.84 283.64,-775.33\"/>\n",
"<text text-anchor=\"middle\" x=\"155.2\" y=\"-547.05\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 12->6 -->\n",
"<g id=\"edge175\" class=\"edge\">\n",
"<title>12->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M56.6,-313.22C106.93,-391.57 331.68,-701.3 404.98,-793.9\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"402.41,-796.3 411.39,-801.92 407.88,-791.93 402.41,-796.3\"/>\n",
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"</g>\n",
"<!-- 12->5 -->\n",
"<g id=\"edge176\" class=\"edge\">\n",
"<title>12->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M61.04,-313.21C130.09,-389.76 426.62,-686.62 522.23,-775.37\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"520.08,-778.15 529.81,-782.35 524.83,-773 520.08,-778.15\"/>\n",
"<text text-anchor=\"middle\" x=\"277.14\" y=\"-548.09\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 12->4 -->\n",
"<g id=\"edge177\" class=\"edge\">\n",
"<title>12->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M67,-313.29C156.35,-384.61 513.73,-644.5 628.31,-721.8\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.5,-724.8 636.76,-727.46 630.4,-718.99 626.5,-724.8\"/>\n",
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"</g>\n",
"<!-- 12->3 -->\n",
"<g id=\"edge178\" class=\"edge\">\n",
"<title>12->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.87,-312.24C183.44,-373.76 591.77,-581.92 716.29,-639.87\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"714.88,-643.07 725.43,-644.08 717.82,-636.72 714.88,-643.07\"/>\n",
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"</g>\n",
"<!-- 12->2 -->\n",
"<g id=\"edge179\" class=\"edge\">\n",
"<title>12->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.84,-306.87C189.46,-349.93 640.13,-496.47 772.67,-534.27\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"771.79,-537.66 782.37,-537 773.69,-530.92 771.79,-537.66\"/>\n",
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"</g>\n",
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"<title>12->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.88,-293.65C183.19,-281.51 590.06,-217.23 715.49,-192.48\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"716.27,-195.9 725.39,-190.49 714.89,-189.03 716.27,-195.9\"/>\n",
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"<!-- 11->20 -->\n",
"<g id=\"edge181\" class=\"edge\">\n",
"<title>11->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M54.35,-417.44C172.26,-404.12 637.15,-330.64 772.41,-304.15\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"773.18,-307.57 782.3,-302.18 771.81,-300.7 773.18,-307.57\"/>\n",
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"<!-- 11->19 -->\n",
"<g id=\"edge182\" class=\"edge\">\n",
"<title>11->19</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M54.15,-421.72C173.77,-427.05 654.01,-427.18 791.93,-422.13\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"792.15,-425.62 802,-421.72 791.87,-418.63 792.15,-425.62\"/>\n",
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"<!-- 11->18 -->\n",
"<g id=\"edge183\" class=\"edge\">\n",
"<title>11->18</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M54.06,-408.51C155.44,-362.45 512.66,-180.63 627.79,-116.77\"/>\n",
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"<!-- 11->17 -->\n",
"<g id=\"edge184\" class=\"edge\">\n",
"<title>11->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M54.15,-403.19C144.25,-343.62 432.5,-134.3 523.94,-62.17\"/>\n",
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"<!-- 11->16 -->\n",
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"<title>11->16</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"409.69,-46.1 413.81,-36.33 404.5,-41.4 409.69,-46.1\"/>\n",
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"<!-- 11->15 -->\n",
"<g id=\"edge186\" class=\"edge\">\n",
"<title>11->15</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"291.95,-66.37 294.42,-56.06 286.06,-62.58 291.95,-66.37\"/>\n",
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"<!-- 11->14 -->\n",
"<g id=\"edge187\" class=\"edge\">\n",
"<title>11->14</title>\n",
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"<g id=\"edge188\" class=\"edge\">\n",
"<title>11->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M36.99,-400.99C54.05,-360.09 86.65,-260.35 99.18,-211.37\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.63,-212 101.6,-201.45 95.83,-210.34 102.63,-212\"/>\n",
"<text text-anchor=\"middle\" x=\"53.59\" y=\"-309.98\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 11->12 -->\n",
"<g id=\"edge189\" class=\"edge\">\n",
"<title>11->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M34.93,-401.03C40.23,-380.68 45.75,-347.33 47.93,-323.49\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"51.43,-323.52 48.68,-313.29 44.45,-323.01 51.43,-323.52\"/>\n",
"<text text-anchor=\"middle\" x=\"26.93\" y=\"-366.06\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 11->11 -->\n",
"<g id=\"edge190\" class=\"edge\">\n",
"<title>11->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M54.24,-425.81C64.02,-425.96 72,-423.71 72,-419.07 72,-416.24 69.04,-414.3 64.51,-413.25\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"64.51,-409.74 54.24,-412.33 63.89,-416.71 64.51,-409.74\"/>\n",
"<text text-anchor=\"middle\" x=\"86.5\" y=\"-415.37\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 11->10 -->\n",
"<g id=\"edge191\" class=\"edge\">\n",
"<title>11->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M24.96,-437.33C26.04,-457.77 31,-491.14 36.29,-514.91\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"32.92,-515.85 38.65,-524.76 39.72,-514.22 32.92,-515.85\"/>\n",
"<text text-anchor=\"middle\" x=\"16.12\" y=\"-479.92\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 11->9 -->\n",
"<g id=\"edge192\" class=\"edge\">\n",
"<title>11->9</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M28.99,-437.17C38.4,-478.22 70.68,-578.54 89.81,-627.31\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"86.58,-628.66 93.56,-636.62 93.07,-626.04 86.58,-628.66\"/>\n",
"<text text-anchor=\"middle\" x=\"66.65\" y=\"-521.04\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 11->8 -->\n",
"<g id=\"edge193\" class=\"edge\">\n",
"<title>11->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M32.78,-437.52C55.64,-492.1 137.14,-652.66 174.13,-716.73\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"171.18,-718.61 179.26,-725.46 177.21,-715.06 171.18,-718.61\"/>\n",
"<text text-anchor=\"middle\" x=\"88.96\" y=\"-580.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 11->7 -->\n",
"<g id=\"edge194\" class=\"edge\">\n",
"<title>11->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M36.52,-437.2C75.56,-499.06 222.38,-701.6 280.44,-774.3\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"278.06,-776.92 287.07,-782.5 283.51,-772.52 278.06,-776.92\"/>\n",
"<text text-anchor=\"middle\" x=\"143.98\" y=\"-609.55\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 11->6 -->\n",
"<g id=\"edge195\" class=\"edge\">\n",
"<title>11->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M41.19,-437.32C99,-502.06 316.69,-720.12 397.61,-794.84\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"395.71,-797.85 405.45,-802.02 400.44,-792.69 395.71,-797.85\"/>\n",
"<text text-anchor=\"middle\" x=\"204.9\" y=\"-619.88\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 11->5 -->\n",
"<g id=\"edge196\" class=\"edge\">\n",
"<title>11->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M46.86,-437.08C125.43,-500.25 414.75,-710.71 516.43,-779.01\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"514.57,-781.97 524.83,-784.61 518.45,-776.15 514.57,-781.97\"/>\n",
"<text text-anchor=\"middle\" x=\"267.14\" y=\"-611.84\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 11->4 -->\n",
"<g id=\"edge197\" class=\"edge\">\n",
"<title>11->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M54.23,-436.33C155.71,-493.52 511.73,-675.02 627.3,-728.64\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"625.92,-731.86 636.47,-732.85 628.84,-725.5 625.92,-731.86\"/>\n",
"<text text-anchor=\"middle\" x=\"326.27\" y=\"-586.28\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 11->3 -->\n",
"<g id=\"edge198\" class=\"edge\">\n",
"<title>11->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M54.13,-430.84C165.5,-472.43 587.67,-609.72 715.59,-646.12\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"714.75,-649.52 725.32,-648.85 716.64,-642.78 714.75,-649.52\"/>\n",
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"</g>\n",
"<!-- 11->2 -->\n",
"<g id=\"edge199\" class=\"edge\">\n",
"<title>11->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M54.43,-426.16C172.36,-450.16 636.47,-523.8 772.1,-540.2\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"772.05,-543.72 782.39,-541.41 772.86,-536.77 772.05,-543.72\"/>\n",
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"</g>\n",
"<!-- 11->1 -->\n",
"<g id=\"edge200\" class=\"edge\">\n",
"<title>11->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M54.1,-413.14C165.4,-382.43 587.5,-245.45 715.52,-198.75\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"717.09,-201.9 725.26,-195.16 714.67,-195.33 717.09,-201.9\"/>\n",
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"</g>\n",
"<!-- 10->20 -->\n",
"<g id=\"edge201\" class=\"edge\">\n",
"<title>10->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.72,-537.02C189.1,-505.07 639.71,-358.83 772.5,-310.48\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"774.04,-313.64 782.22,-306.9 771.63,-307.07 774.04,-313.64\"/>\n",
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"</g>\n",
"<!-- 10->19 -->\n",
"<g id=\"edge202\" class=\"edge\">\n",
"<title>10->19</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.98,-541.38C191.89,-528.06 656.78,-454.58 792.04,-428.09\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"792.81,-431.5 801.93,-426.12 791.44,-424.64 792.81,-431.5\"/>\n",
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"</g>\n",
"<!-- 10->18 -->\n",
"<g id=\"edge203\" class=\"edge\">\n",
"<title>10->18</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.68,-526.93C175.05,-459.54 531.62,-200.57 635.34,-119.16\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"637.81,-121.67 643.48,-112.72 633.47,-116.18 637.81,-121.67\"/>\n",
"<text text-anchor=\"middle\" x=\"340.01\" y=\"-326.85\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 10->17 -->\n",
"<g id=\"edge204\" class=\"edge\">\n",
"<title>10->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M68.77,-524.9C152.5,-448.43 448.54,-152.48 530.75,-63.2\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"533.45,-65.45 537.6,-55.7 528.27,-60.73 533.45,-65.45\"/>\n",
"<text text-anchor=\"middle\" x=\"314.26\" y=\"-282.85\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 10->16 -->\n",
"<g id=\"edge205\" class=\"edge\">\n",
"<title>10->16</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M63.38,-524.71C126.83,-446.18 350.84,-138.02 412.41,-44.81\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"415.48,-46.52 418.01,-36.23 409.61,-42.69 415.48,-46.52\"/>\n",
"<text text-anchor=\"middle\" x=\"252.39\" y=\"-288.56\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 10->15 -->\n",
"<g id=\"edge206\" class=\"edge\">\n",
"<title>10->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M59.07,-524.78C103.59,-448.69 252.67,-156.38 293.76,-65.04\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"297.07,-66.2 297.91,-55.64 290.67,-63.37 297.07,-66.2\"/>\n",
"<text text-anchor=\"middle\" x=\"190.91\" y=\"-298.71\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 10->14 -->\n",
"<g id=\"edge207\" class=\"edge\">\n",
"<title>10->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M55.59,-524.93C83.28,-455.64 164.32,-206.74 186.83,-122.6\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"190.27,-123.3 189.39,-112.74 183.49,-121.54 190.27,-123.3\"/>\n",
"<text text-anchor=\"middle\" x=\"135.71\" y=\"-327.57\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 10->13 -->\n",
"<g id=\"edge208\" class=\"edge\">\n",
"<title>10->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M52.8,-524.68C66.65,-465.99 95.92,-282.42 102.98,-211.59\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"106.47,-211.89 103.89,-201.61 99.5,-211.25 106.47,-211.89\"/>\n",
"<text text-anchor=\"middle\" x=\"92.39\" y=\"-371.94\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 10->12 -->\n",
"<g id=\"edge209\" class=\"edge\">\n",
"<title>10->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M50.37,-524.78C54.12,-481.99 54.37,-374.83 51.12,-323.46\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"54.6,-323.1 50.37,-313.39 47.62,-323.62 54.6,-323.1\"/>\n",
"<text text-anchor=\"middle\" x=\"65.24\" y=\"-427.92\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 10->11 -->\n",
"<g id=\"edge210\" class=\"edge\">\n",
"<title>10->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M48.67,-524.75C47.59,-504.31 42.63,-470.94 37.34,-447.17\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"40.71,-446.23 34.98,-437.31 33.91,-447.85 40.71,-446.23\"/>\n",
"<text text-anchor=\"middle\" x=\"57.51\" y=\"-489.76\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 10->10 -->\n",
"<g id=\"edge211\" class=\"edge\">\n",
"<title>10->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.87,-549.75C83.65,-549.9 91.63,-547.65 91.63,-543.01 91.63,-540.18 88.67,-538.24 84.14,-537.19\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"84.14,-533.68 73.87,-536.26 83.52,-540.65 84.14,-533.68\"/>\n",
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"</g>\n",
"<!-- 10->9 -->\n",
"<g id=\"edge212\" class=\"edge\">\n",
"<title>10->9</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M50.43,-561.25C57.43,-579.41 71.27,-607.41 83.2,-628.07\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"80.22,-629.9 88.35,-636.7 86.24,-626.32 80.22,-629.9\"/>\n",
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"</g>\n",
"<!-- 10->8 -->\n",
"<g id=\"edge213\" class=\"edge\">\n",
"<title>10->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M55.17,-561.36C76.95,-597.21 134.03,-676.35 167.38,-717.56\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"164.86,-720 173.91,-725.51 170.27,-715.56 164.86,-720\"/>\n",
"<text text-anchor=\"middle\" x=\"125.77\" y=\"-628.26\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 10->7 -->\n",
"<g id=\"edge214\" class=\"edge\">\n",
"<title>10->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M59.91,-561.17C99.39,-606.37 216.05,-723.49 272.69,-775.32\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"270.51,-778.06 280.27,-782.18 275.21,-772.87 270.51,-778.06\"/>\n",
"<text text-anchor=\"middle\" x=\"151.8\" y=\"-672.04\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 10->6 -->\n",
"<g id=\"edge215\" class=\"edge\">\n",
"<title>10->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M66,-561.19C126.68,-610.62 313.32,-746.51 392.27,-799.11\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"390.63,-802.22 400.9,-804.81 394.49,-796.38 390.63,-802.22\"/>\n",
"<text text-anchor=\"middle\" x=\"214.64\" y=\"-683.95\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 10->5 -->\n",
"<g id=\"edge216\" class=\"edge\">\n",
"<title>10->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.73,-560.56C158.62,-608.81 419.24,-741.71 515.66,-786.07\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"514.45,-789.36 525,-790.32 517.35,-782.99 514.45,-789.36\"/>\n",
"<text text-anchor=\"middle\" x=\"280.2\" y=\"-677.12\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 10->4 -->\n",
"<g id=\"edge217\" class=\"edge\">\n",
"<title>10->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.82,-555.03C172.71,-592.25 513.36,-703.06 626.77,-735.06\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.14,-738.52 636.71,-737.82 628.01,-731.78 626.14,-738.52\"/>\n",
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"</g>\n",
"<!-- 10->3 -->\n",
"<g id=\"edge218\" class=\"edge\">\n",
"<title>10->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.86,-550.19C183.04,-572.63 589.22,-637.1 715.1,-652.15\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"715.05,-655.66 725.38,-653.34 715.85,-648.71 715.05,-655.66\"/>\n",
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"</g>\n",
"<!-- 10->2 -->\n",
"<g id=\"edge219\" class=\"edge\">\n",
"<title>10->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.78,-545.72C189.17,-550.96 639.02,-551.1 772.18,-546.14\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"772.44,-549.63 782.29,-545.72 772.15,-542.64 772.44,-549.63\"/>\n",
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"<!-- 10->1 -->\n",
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"<title>10->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M73.84,-532.23C183.44,-482.18 592.4,-273.99 716.58,-205.29\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"718.33,-208.33 725.36,-200.4 714.92,-202.21 718.33,-208.33\"/>\n",
"<text text-anchor=\"middle\" x=\"380.71\" y=\"-357.56\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 9->20 -->\n",
"<g id=\"edge221\" class=\"edge\">\n",
"<title>9->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M130.81,-644.04C240.41,-593.99 649.37,-385.79 773.55,-317.1\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"775.29,-320.13 782.33,-312.21 771.89,-314.02 775.29,-320.13\"/>\n",
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"<!-- 9->19 -->\n",
"<g id=\"edge222\" class=\"edge\">\n",
"<title>9->19</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M130.69,-648.88C242,-618.17 664.1,-481.2 792.12,-434.49\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"793.69,-437.64 801.86,-430.9 791.27,-431.08 793.69,-437.64\"/>\n",
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"</g>\n",
"<!-- 9->18 -->\n",
"<g id=\"edge223\" class=\"edge\">\n",
"<title>9->18</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M125.45,-636.79C215.09,-554.67 553.33,-216.53 642.31,-120.24\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"645.04,-122.44 649.21,-112.7 639.88,-117.71 645.04,-122.44\"/>\n",
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"<!-- 9->17 -->\n",
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"<title>9->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M120.1,-636.5C190.86,-548.38 466.9,-168.62 536.23,-64.23\"/>\n",
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"<!-- 9->16 -->\n",
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"<title>9->16</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"420.57,-46.77 421.47,-36.21 414.19,-43.91 420.57,-46.77\"/>\n",
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"<!-- 9->15 -->\n",
"<g id=\"edge226\" class=\"edge\">\n",
"<title>9->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M112.11,-636.79C146.19,-549.62 269.09,-171.86 298.19,-65.55\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"301.58,-66.42 300.78,-55.86 294.82,-64.62 301.58,-66.42\"/>\n",
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"<!-- 9->14 -->\n",
"<g id=\"edge227\" class=\"edge\">\n",
"<title>9->14</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"194.52,-123.16 192.05,-112.86 187.56,-122.46 194.52,-123.16\"/>\n",
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"</g>\n",
"<!-- 9->13 -->\n",
"<g id=\"edge228\" class=\"edge\">\n",
"<title>9->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M106.41,-636.62C111.45,-564.57 111.65,-298.92 107.01,-211.45\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"110.51,-211.24 106.41,-201.47 103.52,-211.66 110.51,-211.24\"/>\n",
"<text text-anchor=\"middle\" x=\"121.21\" y=\"-427.84\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 9->12 -->\n",
"<g id=\"edge229\" class=\"edge\">\n",
"<title>9->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M103.9,-636.73C99.27,-578.33 70.4,-394.47 55.05,-323.48\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"58.41,-322.46 52.8,-313.47 51.58,-324 58.41,-322.46\"/>\n",
"<text text-anchor=\"middle\" x=\"93.98\" y=\"-483.91\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 9->11 -->\n",
"<g id=\"edge230\" class=\"edge\">\n",
"<title>9->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M101.61,-636.72C92.19,-595.66 59.92,-495.35 40.79,-446.57\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"44.02,-445.23 37.03,-437.26 37.53,-447.85 44.02,-445.23\"/>\n",
"<text text-anchor=\"middle\" x=\"85.7\" y=\"-545.44\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 9->10 -->\n",
"<g id=\"edge231\" class=\"edge\">\n",
"<title>9->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M99.8,-636.57C92.8,-618.41 78.95,-590.41 67.02,-569.75\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"70.01,-567.92 61.88,-561.12 63.99,-571.5 70.01,-567.92\"/>\n",
"<text text-anchor=\"middle\" x=\"97.91\" y=\"-606.96\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 9->9 -->\n",
"<g id=\"edge232\" class=\"edge\">\n",
"<title>9->9</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"141.11,-645.48 130.84,-648.07 140.48,-652.45 141.11,-645.48\"/>\n",
"<text text-anchor=\"middle\" x=\"163.1\" y=\"-651.11\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 9->8 -->\n",
"<g id=\"edge233\" class=\"edge\">\n",
"<title>9->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M114.6,-673.12C125.6,-686.31 142.86,-704.16 158.36,-718.62\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"156.05,-721.26 165.79,-725.42 160.77,-716.09 156.05,-721.26\"/>\n",
"<text text-anchor=\"middle\" x=\"121.98\" y=\"-699.67\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 9->7 -->\n",
"<g id=\"edge234\" class=\"edge\">\n",
"<title>9->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M121.42,-673.04C154.41,-700.7 225.33,-752.59 268.5,-780.97\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"266.71,-783.99 277.01,-786.49 270.52,-778.11 266.71,-783.99\"/>\n",
"<text text-anchor=\"middle\" x=\"209.46\" y=\"-715.81\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 9->6 -->\n",
"<g id=\"edge235\" class=\"edge\">\n",
"<title>9->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M130.28,-672.93C188.23,-706.4 325.56,-776.5 391.52,-806.41\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"390.15,-809.64 400.71,-810.52 393.01,-803.25 390.15,-809.64\"/>\n",
"<text text-anchor=\"middle\" x=\"275.4\" y=\"-728.47\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 9->5 -->\n",
"<g id=\"edge236\" class=\"edge\">\n",
"<title>9->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M130.9,-667.37C208.03,-696.85 427.52,-768.29 515.19,-792.61\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"514.4,-796.02 524.96,-795.27 516.23,-789.27 514.4,-796.02\"/>\n",
"<text text-anchor=\"middle\" x=\"308.55\" y=\"-733.79\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 9->4 -->\n",
"<g id=\"edge237\" class=\"edge\">\n",
"<title>9->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M130.89,-662.32C222.79,-681.6 521.16,-728.99 626.54,-741.24\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.37,-744.74 636.7,-742.38 627.15,-737.79 626.37,-744.74\"/>\n",
"<text text-anchor=\"middle\" x=\"364.21\" y=\"-705.58\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 9->3 -->\n",
"<g id=\"edge238\" class=\"edge\">\n",
"<title>9->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M130.66,-657.73C233.11,-662.69 597.39,-662.84 715.37,-658.17\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"715.53,-661.67 725.37,-657.73 715.23,-654.67 715.53,-661.67\"/>\n",
"<text text-anchor=\"middle\" x=\"408.52\" y=\"-661.75\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 9->2 -->\n",
"<g id=\"edge239\" class=\"edge\">\n",
"<title>9->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M130.85,-653.33C240.16,-641.2 647.03,-576.91 772.46,-552.17\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"773.24,-555.58 782.36,-550.17 771.86,-548.72 773.24,-555.58\"/>\n",
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"</g>\n",
"<!-- 9->1 -->\n",
"<g id=\"edge240\" class=\"edge\">\n",
"<title>9->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M130.7,-638.62C235.94,-568.52 616.57,-292.07 724.1,-207.78\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"726.55,-210.3 732.23,-201.36 722.21,-204.81 726.55,-210.3\"/>\n",
"<text text-anchor=\"middle\" x=\"441.9\" y=\"-412\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->20 -->\n",
"<g id=\"edge241\" class=\"edge\">\n",
"<title>8->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.37,-727.47C320.75,-660.07 677.32,-401.11 781.04,-319.69\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"783.51,-322.2 789.18,-313.25 779.16,-316.71 783.51,-322.2\"/>\n",
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"</g>\n",
"<!-- 8->19 -->\n",
"<g id=\"edge242\" class=\"edge\">\n",
"<title>8->19</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.38,-732.98C320.77,-686.92 677.99,-505.1 793.12,-441.24\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"794.9,-444.25 801.93,-436.32 791.49,-438.14 794.9,-444.25\"/>\n",
"<text text-anchor=\"middle\" x=\"491.75\" y=\"-590.91\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->18 -->\n",
"<g id=\"edge243\" class=\"edge\">\n",
"<title>8->18</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M208.74,-725.25C281.82,-634.09 575.92,-229.45 647.93,-121.25\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"650.89,-123.11 653.46,-112.84 645.04,-119.27 650.89,-123.11\"/>\n",
"<text text-anchor=\"middle\" x=\"413.83\" y=\"-427.05\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->17 -->\n",
"<g id=\"edge244\" class=\"edge\">\n",
"<title>8->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M204.27,-725.39C259.6,-629.3 488.37,-180.59 541.21,-64.88\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"544.46,-66.19 545.38,-55.63 538.08,-63.32 544.46,-66.19\"/>\n",
"<text text-anchor=\"middle\" x=\"387.24\" y=\"-398.94\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->16 -->\n",
"<g id=\"edge245\" class=\"edge\">\n",
"<title>8->16</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M200.75,-725.24C238.59,-627.28 388.89,-165.16 421.88,-46.05\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"425.3,-46.79 424.54,-36.22 418.54,-44.96 425.3,-46.79\"/>\n",
"<text text-anchor=\"middle\" x=\"325.81\" y=\"-389.44\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->15 -->\n",
"<g id=\"edge246\" class=\"edge\">\n",
"<title>8->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M197.59,-725.39C218.53,-629.56 289.5,-182.48 302.56,-65.63\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"306.04,-65.98 303.61,-55.67 299.08,-65.25 306.04,-65.98\"/>\n",
"<text text-anchor=\"middle\" x=\"264.58\" y=\"-399.31\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->14 -->\n",
"<g id=\"edge247\" class=\"edge\">\n",
"<title>8->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M194.75,-725.51C200.33,-635.37 200.51,-233.62 195.3,-122.92\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"198.79,-122.65 194.76,-112.85 191.8,-123.02 198.79,-122.65\"/>\n",
"<text text-anchor=\"middle\" x=\"209.52\" y=\"-428.01\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->13 -->\n",
"<g id=\"edge248\" class=\"edge\">\n",
"<title>8->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M192.06,-725.45C184.58,-643.95 132,-310.64 111.26,-211.36\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"114.66,-210.49 109.13,-201.46 107.81,-211.97 114.66,-210.49\"/>\n",
"<text text-anchor=\"middle\" x=\"166.16\" y=\"-472.2\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->12 -->\n",
"<g id=\"edge249\" class=\"edge\">\n",
"<title>8->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M189.38,-725.37C172.03,-655.77 91.13,-406.05 59.23,-322.54\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"62.48,-321.25 55.59,-313.2 55.96,-323.79 62.48,-321.25\"/>\n",
"<text text-anchor=\"middle\" x=\"138.8\" y=\"-527.76\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->11 -->\n",
"<g id=\"edge250\" class=\"edge\">\n",
"<title>8->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M186.54,-725.09C163.69,-670.52 82.18,-509.95 45.19,-445.88\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"48.15,-444.01 40.07,-437.15 42.11,-447.55 48.15,-444.01\"/>\n",
"<text text-anchor=\"middle\" x=\"130.37\" y=\"-589.29\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->10 -->\n",
"<g id=\"edge251\" class=\"edge\">\n",
"<title>8->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M183.79,-725.19C162,-689.34 104.93,-610.2 71.58,-569\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"74.1,-566.55 65.04,-561.05 68.69,-570.99 74.1,-566.55\"/>\n",
"<text text-anchor=\"middle\" x=\"142.18\" y=\"-635.89\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->9 -->\n",
"<g id=\"edge252\" class=\"edge\">\n",
"<title>8->9</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M181.33,-725.24C170.33,-712.04 153.07,-694.2 137.57,-679.73\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"139.87,-677.1 130.13,-672.94 135.15,-682.27 139.87,-677.1\"/>\n",
"<text text-anchor=\"middle\" x=\"173.95\" y=\"-706.29\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->8 -->\n",
"<g id=\"edge253\" class=\"edge\">\n",
"<title>8->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.57,-750.29C229.35,-750.43 237.33,-748.18 237.33,-743.54 237.33,-740.72 234.37,-738.78 229.84,-737.73\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.84,-734.21 219.57,-736.8 229.21,-741.18 229.84,-734.21\"/>\n",
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"</g>\n",
"<!-- 8->7 -->\n",
"<g id=\"edge254\" class=\"edge\">\n",
"<title>8->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M215.08,-761.6C230.23,-770.59 250.42,-781.06 267.76,-788.99\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"266.48,-792.25 277.04,-793.1 269.32,-785.85 266.48,-792.25\"/>\n",
"<text text-anchor=\"middle\" x=\"226.92\" y=\"-779.09\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->6 -->\n",
"<g id=\"edge255\" class=\"edge\">\n",
"<title>8->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.35,-757.14C261.62,-773.77 343.46,-800.5 391.14,-813.39\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"390.5,-816.83 401.06,-815.99 392.27,-810.06 390.5,-816.83\"/>\n",
"<text text-anchor=\"middle\" x=\"319.74\" y=\"-774.06\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->5 -->\n",
"<g id=\"edge256\" class=\"edge\">\n",
"<title>8->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.36,-751.68C282.9,-765.55 441.84,-790.88 514.59,-798.93\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"514.37,-802.42 524.68,-799.99 515.1,-795.46 514.37,-802.42\"/>\n",
"<text text-anchor=\"middle\" x=\"352.48\" y=\"-779.1\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 8->4 -->\n",
"<g id=\"edge257\" class=\"edge\">\n",
"<title>8->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.37,-746.9C299.27,-751.25 534.41,-751.42 626.36,-747.41\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.77,-750.89 636.59,-746.91 626.43,-743.9 626.77,-750.89\"/>\n",
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"<!-- 8->3 -->\n",
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"<title>8->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.45,-742.38C311.2,-732.61 610.03,-685.45 715.37,-664.35\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"716.15,-667.76 725.24,-662.33 714.74,-660.9 716.15,-667.76\"/>\n",
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"</g>\n",
"<!-- 8->2 -->\n",
"<g id=\"edge259\" class=\"edge\">\n",
"<title>8->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M219.36,-737.84C318.09,-710.92 659.25,-600.26 772.59,-558.65\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"774.06,-561.83 782.22,-555.07 771.62,-555.27 774.06,-561.83\"/>\n",
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"<g id=\"edge260\" class=\"edge\">\n",
"<title>8->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M214.18,-725.52C303.82,-643.4 642.06,-305.26 731.04,-208.97\"/>\n",
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"<title>7->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M326.27,-782.41C410,-705.93 706.05,-409.98 788.26,-320.71\"/>\n",
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"<!-- 7->19 -->\n",
"<g id=\"edge262\" class=\"edge\">\n",
"<title>7->19</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M331.28,-784.63C421.38,-725.06 709.63,-515.74 801.07,-443.61\"/>\n",
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"<!-- 7->18 -->\n",
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"<title>7->18</title>\n",
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"<title>7->17</title>\n",
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"<title>7->16</title>\n",
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"<!-- 7->15 -->\n",
"<g id=\"edge266\" class=\"edge\">\n",
"<title>7->15</title>\n",
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"</g>\n",
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"<title>7->14</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"203.05,-121.69 197.56,-112.63 196.2,-123.14 203.05,-121.69\"/>\n",
"<text text-anchor=\"middle\" x=\"266.13\" y=\"-441.27\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 7->13 -->\n",
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"<title>7->13</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"118.96,-209.45 112.12,-201.36 112.42,-211.96 118.96,-209.45\"/>\n",
"<text text-anchor=\"middle\" x=\"222.79\" y=\"-485.45\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>7->12</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"67.12,-320.31 59.1,-313.39 61.05,-323.81 67.12,-320.31\"/>\n",
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"</g>\n",
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"<title>7->11</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"53.07,-442.66 44.07,-437.08 47.63,-447.06 53.07,-442.66\"/>\n",
"<text text-anchor=\"middle\" x=\"187.15\" y=\"-602.64\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 7->10 -->\n",
"<g id=\"edge271\" class=\"edge\">\n",
"<title>7->10</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"80.26,-565.46 70.5,-561.34 75.56,-570.65 80.26,-565.46\"/>\n",
"<text text-anchor=\"middle\" x=\"198.96\" y=\"-664.08\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 7->9 -->\n",
"<g id=\"edge272\" class=\"edge\">\n",
"<title>7->9</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"141.02,-671.34 130.72,-668.83 137.21,-677.21 141.02,-671.34\"/>\n",
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"</g>\n",
"<!-- 7->8 -->\n",
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"<title>7->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M281.38,-782.45C266.23,-773.47 246.04,-763 228.7,-755.07\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.98,-751.8 219.42,-750.95 227.14,-758.2 229.98,-751.8\"/>\n",
"<text text-anchor=\"middle\" x=\"240.54\" y=\"-757.56\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 7->7 -->\n",
"<g id=\"edge274\" class=\"edge\">\n",
"<title>7->7</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.65,-791.18 331.37,-793.77 341.02,-798.15 341.65,-791.18\"/>\n",
"<text text-anchor=\"middle\" x=\"363.63\" y=\"-796.81\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 7->6 -->\n",
"<g id=\"edge275\" class=\"edge\">\n",
"<title>7->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M331.36,-810.66C348.77,-814.57 371.6,-818.34 390.7,-820.51\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"390.57,-824.01 400.87,-821.54 391.28,-817.05 390.57,-824.01\"/>\n",
"<text text-anchor=\"middle\" x=\"346.53\" y=\"-819.38\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 7->5 -->\n",
"<g id=\"edge276\" class=\"edge\">\n",
"<title>7->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M331.61,-804.94C376.21,-807.83 464.42,-808.02 514.67,-805.5\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"515,-808.99 524.79,-804.93 514.61,-802 515,-808.99\"/>\n",
"<text text-anchor=\"middle\" x=\"408.64\" y=\"-794.02\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 7->4 -->\n",
"<g id=\"edge277\" class=\"edge\">\n",
"<title>7->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M331.17,-800.01C394.69,-793.8 553.59,-768.84 626.36,-753.88\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"627.39,-757.24 636.45,-751.75 625.94,-750.4 627.39,-757.24\"/>\n",
"<text text-anchor=\"middle\" x=\"464.26\" y=\"-780.75\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 7->3 -->\n",
"<g id=\"edge278\" class=\"edge\">\n",
"<title>7->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M331.22,-795.26C408.23,-774.75 628.41,-703.42 715.99,-670.86\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"717.38,-674.07 725.51,-667.27 714.91,-667.52 717.38,-674.07\"/>\n",
"<text text-anchor=\"middle\" x=\"509.11\" y=\"-736.86\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 7->2 -->\n",
"<g id=\"edge279\" class=\"edge\">\n",
"<title>7->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M331.34,-790.23C416.41,-751.99 677.3,-619.27 773.45,-565.6\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"775.24,-568.62 782.23,-560.66 771.8,-562.52 775.24,-568.62\"/>\n",
"<text text-anchor=\"middle\" x=\"537.9\" y=\"-681.72\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 7->1 -->\n",
"<g id=\"edge280\" class=\"edge\">\n",
"<title>7->1</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"739.72,-211.8 742.29,-201.52 733.87,-207.96 739.72,-211.8\"/>\n",
"<text text-anchor=\"middle\" x=\"543.2\" y=\"-499.86\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 6->20 -->\n",
"<g id=\"edge281\" class=\"edge\">\n",
"<title>6->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M444.82,-801.84C508.27,-723.31 732.28,-415.15 793.85,-321.95\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"796.92,-323.65 799.45,-313.37 791.06,-319.83 796.92,-323.65\"/>\n",
"<text text-anchor=\"middle\" x=\"633.83\" y=\"-565.69\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 6->19 -->\n",
"<g id=\"edge282\" class=\"edge\">\n",
"<title>6->19</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"810.76,-447.17 814.88,-437.4 805.57,-442.47 810.76,-447.17\"/>\n",
"<text text-anchor=\"middle\" x=\"643.85\" y=\"-627.31\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 6->18 -->\n",
"<g id=\"edge283\" class=\"edge\">\n",
"<title>6->18</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M436.5,-801.84C474.33,-703.87 624.63,-241.75 657.62,-122.65\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"661.04,-123.39 660.28,-112.82 654.29,-121.56 661.04,-123.39\"/>\n",
"<text text-anchor=\"middle\" x=\"561.56\" y=\"-466.04\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 6->17 -->\n",
"<g id=\"edge284\" class=\"edge\">\n",
"<title>6->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M433.24,-801.92C455.45,-699.05 536.05,-191.07 550.3,-65.66\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"553.79,-66 551.38,-55.69 546.83,-65.25 553.79,-66\"/>\n",
"<text text-anchor=\"middle\" x=\"477.27\" y=\"-422.59\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 6->16 -->\n",
"<g id=\"edge285\" class=\"edge\">\n",
"<title>6->16</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M430.28,-801.88C436.16,-697.31 436.33,-174.6 430.78,-46.42\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"434.26,-46.05 430.28,-36.23 427.27,-46.39 434.26,-46.05\"/>\n",
"<text text-anchor=\"middle\" x=\"445.03\" y=\"-427.95\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 6->15 -->\n",
"<g id=\"edge286\" class=\"edge\">\n",
"<title>6->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M427.43,-801.98C417.11,-699.27 336.84,-191.31 311.36,-65.74\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"314.75,-64.83 309.28,-55.75 307.89,-66.26 314.75,-64.83\"/>\n",
"<text text-anchor=\"middle\" x=\"383.9\" y=\"-437.66\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 6->14 -->\n",
"<g id=\"edge287\" class=\"edge\">\n",
"<title>6->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M424.52,-801.86C398.75,-703.88 248.54,-240.96 204.33,-122.34\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"207.55,-120.97 200.74,-112.86 201.01,-123.45 207.55,-120.97\"/>\n",
"<text text-anchor=\"middle\" x=\"328.92\" y=\"-465.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 6->13 -->\n",
"<g id=\"edge288\" class=\"edge\">\n",
"<title>6->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M421.51,-802.03C382.2,-712.5 181.25,-317.67 120.72,-210.33\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"123.63,-208.36 115.63,-201.41 117.55,-211.83 123.63,-208.36\"/>\n",
"<text text-anchor=\"middle\" x=\"285.62\" y=\"-509.98\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 6->12 -->\n",
"<g id=\"edge289\" class=\"edge\">\n",
"<title>6->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M418.1,-802.05C367.77,-723.7 143.02,-413.97 69.72,-321.37\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"72.29,-318.98 63.31,-313.35 66.82,-323.35 72.29,-318.98\"/>\n",
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"</g>\n",
"<!-- 6->11 -->\n",
"<g id=\"edge290\" class=\"edge\">\n",
"<title>6->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M413.88,-801.89C356.07,-737.15 138.38,-519.1 57.46,-444.37\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"59.36,-441.36 49.62,-437.19 54.63,-446.53 59.36,-441.36\"/>\n",
"<text text-anchor=\"middle\" x=\"250.17\" y=\"-626.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 6->10 -->\n",
"<g id=\"edge291\" class=\"edge\">\n",
"<title>6->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M408.7,-801.96C348.02,-752.53 161.38,-616.64 82.43,-564.04\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"84.07,-560.93 73.8,-558.34 80.21,-566.77 84.07,-560.93\"/>\n",
"<text text-anchor=\"middle\" x=\"245.56\" y=\"-671.8\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 6->9 -->\n",
"<g id=\"edge292\" class=\"edge\">\n",
"<title>6->9</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M401.39,-802.02C343.44,-768.55 206.11,-698.46 140.15,-668.54\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"141.52,-665.32 130.96,-664.43 138.66,-671.71 141.52,-665.32\"/>\n",
"<text text-anchor=\"middle\" x=\"256.27\" y=\"-724.08\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 6->8 -->\n",
"<g id=\"edge293\" class=\"edge\">\n",
"<title>6->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M401.05,-806.54C358.77,-789.92 276.94,-763.19 229.26,-750.3\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.9,-746.85 219.34,-747.69 228.12,-753.62 229.9,-746.85\"/>\n",
"<text text-anchor=\"middle\" x=\"300.65\" y=\"-767.22\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 6->7 -->\n",
"<g id=\"edge294\" class=\"edge\">\n",
"<title>6->7</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.63,-796.64 331.33,-799.11 340.93,-803.6 341.63,-796.64\"/>\n",
"<text text-anchor=\"middle\" x=\"385.68\" y=\"-808.87\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 6->6 -->\n",
"<g id=\"edge295\" class=\"edge\">\n",
"<title>6->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M455.31,-826.88C465.09,-827.03 473.07,-824.78 473.07,-820.14 473.07,-817.31 470.11,-815.37 465.58,-814.32\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.59,-810.81 455.31,-813.4 464.96,-817.78 465.59,-810.81\"/>\n",
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"</g>\n",
"<!-- 6->5 -->\n",
"<g id=\"edge296\" class=\"edge\">\n",
"<title>6->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M455.24,-821.54C472.77,-820.01 495.81,-816.58 515.01,-812.73\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"515.88,-816.12 524.94,-810.62 514.43,-809.27 515.88,-816.12\"/>\n",
"<text text-anchor=\"middle\" x=\"470.62\" y=\"-805.94\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 6->4 -->\n",
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"<title>6->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M455.11,-815.98C497.44,-805.24 579.39,-778.87 627.04,-760.9\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"628.58,-764.05 636.66,-757.2 626.07,-757.52 628.58,-764.05\"/>\n",
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"<title>6->3</title>\n",
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"<title>6->2</title>\n",
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"<title>6->1</title>\n",
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"<title>5->20</title>\n",
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"<title>5->19</title>\n",
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"<title>5->18</title>\n",
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"<title>5->17</title>\n",
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"<!-- 5->16 -->\n",
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"<title>5->16</title>\n",
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"<title>5->15</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"319.32,-64.08 312.51,-55.96 312.77,-66.55 319.32,-64.08\"/>\n",
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"<!-- 5->14 -->\n",
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"<title>5->14</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"212.2,-119.61 204.21,-112.65 206.11,-123.07 212.2,-119.61\"/>\n",
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"<title>5->13</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"128.82,-206.98 119.85,-201.33 123.34,-211.33 128.82,-206.98\"/>\n",
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"<title>5->12</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"78.55,-317.49 68.83,-313.29 73.81,-322.64 78.55,-317.49\"/>\n",
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"<title>5->11</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"64.44,-437.61 54.18,-434.97 60.56,-443.43 64.44,-437.61\"/>\n",
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"</g>\n",
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"<g id=\"edge311\" class=\"edge\">\n",
"<title>5->10</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"84.18,-554.16 73.63,-553.2 81.29,-560.53 84.18,-554.16\"/>\n",
"<text text-anchor=\"middle\" x=\"289.44\" y=\"-659\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 5->9 -->\n",
"<g id=\"edge312\" class=\"edge\">\n",
"<title>5->9</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"141.21,-659.3 130.64,-660.05 139.37,-666.06 141.21,-659.3\"/>\n",
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"</g>\n",
"<!-- 5->8 -->\n",
"<g id=\"edge313\" class=\"edge\">\n",
"<title>5->8</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.96,-741.63 219.65,-744.07 229.23,-748.59 229.96,-741.63\"/>\n",
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"<!-- 5->7 -->\n",
"<g id=\"edge314\" class=\"edge\">\n",
"<title>5->7</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.14,-792.03 331.35,-796.1 341.54,-799.02 341.14,-792.03\"/>\n",
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"</g>\n",
"<!-- 5->6 -->\n",
"<g id=\"edge315\" class=\"edge\">\n",
"<title>5->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M524.84,-799.11C507.3,-800.64 484.26,-804.08 465.07,-807.92\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"464.19,-804.53 455.14,-810.03 465.65,-811.38 464.19,-804.53\"/>\n",
"<text text-anchor=\"middle\" x=\"509.45\" y=\"-792.32\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 5->5 -->\n",
"<g id=\"edge316\" class=\"edge\">\n",
"<title>5->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M579.25,-807.25C589.03,-807.4 597.01,-805.15 597.01,-800.51 597.01,-797.68 594.05,-795.74 589.52,-794.69\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"589.52,-791.18 579.25,-793.77 588.89,-798.15 589.52,-791.18\"/>\n",
"<text text-anchor=\"middle\" x=\"611.51\" y=\"-796.81\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 5->4 -->\n",
"<g id=\"edge317\" class=\"edge\">\n",
"<title>5->4</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"633.87,-769.78 640.79,-761.76 630.38,-763.71 633.87,-769.78\"/>\n",
"<text text-anchor=\"middle\" x=\"591.03\" y=\"-768.77\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 5->3 -->\n",
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"<title>5->3</title>\n",
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"<text text-anchor=\"middle\" x=\"638.54\" y=\"-721.79\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 5->2 -->\n",
"<g id=\"edge319\" class=\"edge\">\n",
"<title>5->2</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"792.33,-570.83 796.35,-561.02 787.1,-566.18 792.33,-570.83\"/>\n",
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"</g>\n",
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"<g id=\"edge320\" class=\"edge\">\n",
"<title>5->1</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"749.99,-212.12 749.19,-201.55 743.23,-210.31 749.99,-212.12\"/>\n",
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"<g id=\"edge321\" class=\"edge\">\n",
"<title>4->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M672.77,-725.47C700.46,-656.17 781.5,-407.27 804.01,-323.14\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"807.45,-323.83 806.57,-313.27 800.67,-322.07 807.45,-323.83\"/>\n",
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"</g>\n",
"<!-- 4->19 -->\n",
"<g id=\"edge322\" class=\"edge\">\n",
"<title>4->19</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M676.98,-725.3C709.32,-671.2 790.78,-511.64 819.34,-446.85\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"822.71,-447.86 823.45,-437.3 816.28,-445.1 822.71,-447.86\"/>\n",
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"</g>\n",
"<!-- 4->18 -->\n",
"<g id=\"edge323\" class=\"edge\">\n",
"<title>4->18</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"670.28,-122.65 666.25,-112.85 663.29,-123.02 670.28,-122.65\"/>\n",
"<text text-anchor=\"middle\" x=\"681.01\" y=\"-428.01\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 4->17 -->\n",
"<g id=\"edge324\" class=\"edge\">\n",
"<title>4->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M663.28,-725.38C653.89,-629.51 583.27,-182.44 559.35,-65.62\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"562.73,-64.73 557.24,-55.66 555.88,-66.17 562.73,-64.73\"/>\n",
"<text text-anchor=\"middle\" x=\"625.81\" y=\"-399.3\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 4->16 -->\n",
"<g id=\"edge325\" class=\"edge\">\n",
"<title>4->16</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M660.26,-725.27C634.5,-627.28 484.29,-164.37 440.07,-45.74\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"443.3,-44.38 436.48,-36.26 436.75,-46.86 443.3,-44.38\"/>\n",
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"</g>\n",
"<!-- 4->15 -->\n",
"<g id=\"edge326\" class=\"edge\">\n",
"<title>4->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M657.13,-725.41C614.63,-629.3 385.84,-179.84 321.08,-64.59\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"324,-62.65 316.02,-55.68 317.92,-66.1 324,-62.65\"/>\n",
"<text text-anchor=\"middle\" x=\"496.35\" y=\"-383.8\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 4->14 -->\n",
"<g id=\"edge327\" class=\"edge\">\n",
"<title>4->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M653.58,-725.48C594.59,-634.64 299.83,-228.6 215.07,-120.87\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"217.62,-118.45 208.66,-112.79 212.13,-122.8 217.62,-118.45\"/>\n",
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"</g>\n",
"<!-- 4->13 -->\n",
"<g id=\"edge328\" class=\"edge\">\n",
"<title>4->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M649.11,-725.32C574.23,-642.72 235.97,-304.15 133.23,-208.53\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"135.4,-205.78 125.68,-201.56 130.65,-210.92 135.4,-205.78\"/>\n",
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"<!-- 4->12 -->\n",
"<g id=\"edge329\" class=\"edge\">\n",
"<title>4->12</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"83.94,-313.87 73.69,-311.21 80.05,-319.69 83.94,-313.87\"/>\n",
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"</g>\n",
"<!-- 4->11 -->\n",
"<g id=\"edge330\" class=\"edge\">\n",
"<title>4->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M636.58,-726.29C535.11,-669.1 179.08,-487.59 63.51,-433.98\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"64.89,-430.76 54.34,-429.76 61.97,-437.12 64.89,-430.76\"/>\n",
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"</g>\n",
"<!-- 4->10 -->\n",
"<g id=\"edge331\" class=\"edge\">\n",
"<title>4->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M636.63,-731.52C537.73,-694.3 197.08,-583.5 83.68,-551.49\"/>\n",
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"<!-- 4->9 -->\n",
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"<title>4->9</title>\n",
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"<!-- 4->8 -->\n",
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"<title>4->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M636.77,-740.19C556.87,-735.84 321.73,-735.67 229.78,-739.68\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.37,-736.2 219.55,-740.18 229.71,-743.19 229.37,-736.2\"/>\n",
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"<!-- 4->7 -->\n",
"<g id=\"edge334\" class=\"edge\">\n",
"<title>4->7</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"340.56,-786.81 331.5,-792.3 342.01,-793.66 340.56,-786.81\"/>\n",
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"<!-- 4->6 -->\n",
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"<title>4->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M636.78,-747.7C594.45,-758.44 512.5,-784.81 464.84,-802.79\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"463.3,-799.63 455.22,-806.49 465.81,-806.17 463.3,-799.63\"/>\n",
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"<!-- 4->5 -->\n",
"<g id=\"edge336\" class=\"edge\">\n",
"<title>4->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M636.69,-750.97C620.76,-757.78 600.57,-767.85 583.89,-777.2\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"581.95,-774.27 575.03,-782.29 585.44,-780.34 581.95,-774.27\"/>\n",
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"<!-- 4->4 -->\n",
"<g id=\"edge337\" class=\"edge\">\n",
"<title>4->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M691.05,-750.29C700.84,-750.43 708.81,-748.18 708.81,-743.54 708.81,-740.72 705.85,-738.78 701.32,-737.73\"/>\n",
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"<title>4->3</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"737.72,-682.91 741.6,-673.05 732.42,-678.33 737.72,-682.91\"/>\n",
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"<title>4->2</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"798.85,-571.37 801.17,-561.03 792.91,-567.67 798.85,-571.37\"/>\n",
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"<title>4->1</title>\n",
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"<title>3->20</title>\n",
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"<title>3->19</title>\n",
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"<title>3->18</title>\n",
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"<title>3->17</title>\n",
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"<title>3->16</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"448.1,-43.03 440.11,-36.08 442.02,-46.5 448.1,-43.03\"/>\n",
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"</g>\n",
"<!-- 3->15 -->\n",
"<g id=\"edge346\" class=\"edge\">\n",
"<title>3->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M742.26,-636.57C685.14,-548.45 408.51,-167.35 326.81,-63.7\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"329.35,-61.28 320.39,-55.63 323.87,-65.64 329.35,-61.28\"/>\n",
"<text text-anchor=\"middle\" x=\"549.03\" y=\"-353.94\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3->14 -->\n",
"<g id=\"edge347\" class=\"edge\">\n",
"<title>3->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M737.84,-636.59C662.96,-553.99 324.69,-215.42 221.96,-119.8\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"224.13,-117.05 214.41,-112.83 219.38,-122.19 224.13,-117.05\"/>\n",
"<text text-anchor=\"middle\" x=\"494.4\" y=\"-367\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3->13 -->\n",
"<g id=\"edge348\" class=\"edge\">\n",
"<title>3->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M732.14,-636.71C639.44,-562.85 257.91,-285.42 139.07,-205.14\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"140.91,-202.15 130.65,-199.49 137.01,-207.96 140.91,-202.15\"/>\n",
"<text text-anchor=\"middle\" x=\"450.11\" y=\"-409.72\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3->12 -->\n",
"<g id=\"edge349\" class=\"edge\">\n",
"<title>3->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.3,-637.71C615.74,-576.18 207.4,-368.02 82.89,-310.08\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"84.29,-306.87 73.74,-305.86 81.36,-313.23 84.29,-306.87\"/>\n",
"<text text-anchor=\"middle\" x=\"418.59\" y=\"-477.69\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3->11 -->\n",
"<g id=\"edge350\" class=\"edge\">\n",
"<title>3->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.42,-643.04C614.04,-601.45 191.88,-464.17 63.95,-427.77\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"64.79,-424.37 54.22,-425.04 62.9,-431.11 64.79,-424.37\"/>\n",
"<text text-anchor=\"middle\" x=\"409.18\" y=\"-524.2\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3->10 -->\n",
"<g id=\"edge351\" class=\"edge\">\n",
"<title>3->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.31,-647.63C616.14,-625.19 209.95,-560.72 84.07,-545.67\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"84.13,-542.16 73.79,-544.49 83.32,-549.11 84.13,-542.16\"/>\n",
"<text text-anchor=\"middle\" x=\"419.19\" y=\"-585.45\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3->9 -->\n",
"<g id=\"edge352\" class=\"edge\">\n",
"<title>3->9</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.49,-651.9C623.03,-646.94 258.75,-646.79 140.77,-651.46\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"140.61,-647.96 130.77,-651.89 140.91,-654.96 140.61,-647.96\"/>\n",
"<text text-anchor=\"middle\" x=\"418.63\" y=\"-640.48\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3->8 -->\n",
"<g id=\"edge353\" class=\"edge\">\n",
"<title>3->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.42,-655.98C633.67,-665.75 334.84,-712.91 229.5,-734.01\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"228.73,-730.6 219.63,-736.03 230.13,-737.45 228.73,-730.6\"/>\n",
"<text text-anchor=\"middle\" x=\"462.96\" y=\"-683.8\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3->7 -->\n",
"<g id=\"edge354\" class=\"edge\">\n",
"<title>3->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.45,-660.07C648.44,-680.57 428.26,-751.91 340.69,-784.47\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"339.29,-781.26 331.17,-788.06 341.76,-787.81 339.29,-781.26\"/>\n",
"<text text-anchor=\"middle\" x=\"518.57\" y=\"-711.07\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3->6 -->\n",
"<g id=\"edge355\" class=\"edge\">\n",
"<title>3->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.34,-664.36C666.75,-690.13 528.83,-760.16 463.51,-797.02\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"461.75,-794 454.79,-802 465.21,-800.08 461.75,-794\"/>\n",
"<text text-anchor=\"middle\" x=\"579.93\" y=\"-719.49\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3->5 -->\n",
"<g id=\"edge356\" class=\"edge\">\n",
"<title>3->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.24,-668.95C686.38,-693.78 615.79,-744.95 577.79,-775.73\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"575.25,-773.29 569.75,-782.34 579.7,-778.69 575.25,-773.29\"/>\n",
"<text text-anchor=\"middle\" x=\"666.01\" y=\"-711.14\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3->4 -->\n",
"<g id=\"edge357\" class=\"edge\">\n",
"<title>3->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.94,-672.99C711.49,-685.92 694.08,-703.35 681.32,-717.7\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"678.64,-715.45 674.76,-725.31 683.94,-720.02 678.64,-715.45\"/>\n",
"<text text-anchor=\"middle\" x=\"718.13\" y=\"-684.15\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3->3 -->\n",
"<g id=\"edge358\" class=\"edge\">\n",
"<title>3->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M779.78,-661.56C789.57,-661.7 797.54,-659.45 797.54,-654.81 797.54,-651.99 794.58,-650.05 790.05,-649\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"790.06,-645.48 779.78,-648.07 789.43,-652.45 790.06,-645.48\"/>\n",
"<text text-anchor=\"middle\" x=\"812.04\" y=\"-651.11\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3->2 -->\n",
"<g id=\"edge359\" class=\"edge\">\n",
"<title>3->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M767.77,-636.74C778.79,-618.88 792.94,-591.39 801.82,-570.77\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"805.15,-571.85 805.71,-561.27 798.68,-569.19 805.15,-571.85\"/>\n",
"<text text-anchor=\"middle\" x=\"770.29\" y=\"-607.55\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 3->1 -->\n",
"<g id=\"edge360\" class=\"edge\">\n",
"<title>3->1</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"759.45,-211.24 755.35,-201.47 752.46,-211.66 759.45,-211.24\"/>\n",
"<text text-anchor=\"middle\" x=\"741.16\" y=\"-427.84\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 2->20 -->\n",
"<g id=\"edge361\" class=\"edge\">\n",
"<title>2->20</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M813.25,-524.78C817,-481.99 817.25,-374.83 814,-323.46\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"817.48,-323.1 813.25,-313.39 810.5,-323.62 817.48,-323.1\"/>\n",
"<text text-anchor=\"middle\" x=\"799.12\" y=\"-427.92\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 2->19 -->\n",
"<g id=\"edge362\" class=\"edge\">\n",
"<title>2->19</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M817.44,-524.97C822.74,-504.62 828.26,-471.27 830.44,-447.43\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"833.95,-447.46 831.19,-437.23 826.96,-446.94 833.95,-447.46\"/>\n",
"<text text-anchor=\"middle\" x=\"838.44\" y=\"-490\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 2->18 -->\n",
"<g id=\"edge363\" class=\"edge\">\n",
"<title>2->18</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M806.56,-524.84C789.21,-455.24 708.32,-205.51 676.41,-122\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"679.66,-120.71 672.77,-112.67 673.14,-123.26 679.66,-120.71\"/>\n",
"<text text-anchor=\"middle\" x=\"755.99\" y=\"-327.22\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 2->17 -->\n",
"<g id=\"edge364\" class=\"edge\">\n",
"<title>2->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M803.29,-525C770.54,-449.17 621.39,-155.93 569.6,-64.79\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"572.5,-62.81 564.47,-55.89 566.43,-66.3 572.5,-62.81\"/>\n",
"<text text-anchor=\"middle\" x=\"700.94\" y=\"-298.69\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 2->16 -->\n",
"<g id=\"edge365\" class=\"edge\">\n",
"<title>2->16</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M799.54,-524.92C749.21,-446.57 524.46,-136.84 451.16,-44.24\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"453.73,-41.84 444.75,-36.22 448.26,-46.21 453.73,-41.84\"/>\n",
"<text text-anchor=\"middle\" x=\"639.85\" y=\"-273.38\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 2->15 -->\n",
"<g id=\"edge366\" class=\"edge\">\n",
"<title>2->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M795.1,-524.94C726.05,-448.38 429.52,-151.52 333.91,-62.77\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"336.06,-59.99 326.33,-55.79 331.31,-65.14 336.06,-59.99\"/>\n",
"<text text-anchor=\"middle\" x=\"579\" y=\"-297.65\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 2->14 -->\n",
"<g id=\"edge367\" class=\"edge\">\n",
"<title>2->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M789.14,-524.85C699.79,-453.53 342.41,-193.64 227.84,-116.34\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.64,-113.34 219.38,-110.68 225.74,-119.15 229.64,-113.34\"/>\n",
"<text text-anchor=\"middle\" x=\"522.99\" y=\"-324.4\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 2->13 -->\n",
"<g id=\"edge368\" class=\"edge\">\n",
"<title>2->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M782.27,-525.9C672.71,-464.38 264.37,-256.22 139.85,-198.28\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"141.26,-195.07 130.71,-194.06 138.33,-201.42 141.26,-195.07\"/>\n",
"<text text-anchor=\"middle\" x=\"468.31\" y=\"-350.89\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 2->12 -->\n",
"<g id=\"edge369\" class=\"edge\">\n",
"<title>2->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M782.31,-531.27C666.68,-488.22 216.01,-341.67 83.47,-303.87\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"84.35,-300.48 73.78,-301.14 82.45,-307.22 84.35,-300.48\"/>\n",
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"<!-- 2->11 -->\n",
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"<title>2->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M782.09,-535.92C664.15,-511.92 200.04,-438.28 64.41,-421.88\"/>\n",
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"<text text-anchor=\"middle\" x=\"437.75\" y=\"-467.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<!-- 2->10 -->\n",
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"<title>2->10</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"83.7,-536.39 73.85,-540.29 83.99,-543.38 83.7,-536.39\"/>\n",
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"<!-- 2->9 -->\n",
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"<title>2->9</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"139.87,-642.24 130.75,-647.65 141.25,-649.11 139.87,-642.24\"/>\n",
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"<!-- 2->8 -->\n",
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"<title>2->8</title>\n",
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"<!-- 2->7 -->\n",
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"<title>2->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M782.31,-553.29C697.23,-591.53 436.35,-724.24 340.19,-777.91\"/>\n",
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"<!-- 2->6 -->\n",
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"<title>2->6</title>\n",
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"<!-- 2->5 -->\n",
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"<title>2->5</title>\n",
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"<title>2->4</title>\n",
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"<title>2->3</title>\n",
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"<title>2->2</title>\n",
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"<title>2->1</title>\n",
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"<title>1->20</title>\n",
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"<title>1->19</title>\n",
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"<!-- 1->18 -->\n",
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"<title>1->18</title>\n",
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"<title>1->17</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M734.72,-165.1C701.73,-137.44 630.81,-85.56 587.64,-57.17\"/>\n",
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"<text text-anchor=\"middle\" x=\"675.68\" y=\"-114.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->16 -->\n",
"<g id=\"edge385\" class=\"edge\">\n",
"<title>1->16</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.86,-165.21C667.91,-131.74 530.58,-61.64 464.63,-31.73\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.99,-28.5 455.43,-27.62 463.13,-34.89 465.99,-28.5\"/>\n",
"<text text-anchor=\"middle\" x=\"609.74\" y=\"-87.27\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->15 -->\n",
"<g id=\"edge386\" class=\"edge\">\n",
"<title>1->15</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.24,-170.77C648.11,-141.29 428.62,-69.85 340.95,-45.53\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.74,-42.12 331.18,-42.87 339.91,-48.87 341.74,-42.12\"/>\n",
"<text text-anchor=\"middle\" x=\"518.59\" y=\"-96.95\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->14 -->\n",
"<g id=\"edge387\" class=\"edge\">\n",
"<title>1->14</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.25,-175.82C633.35,-156.54 334.98,-109.15 229.6,-96.9\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.77,-93.4 219.45,-95.76 228.99,-100.35 229.77,-93.4\"/>\n",
"<text text-anchor=\"middle\" x=\"462.93\" y=\"-125.16\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->13 -->\n",
"<g id=\"edge388\" class=\"edge\">\n",
"<title>1->13</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.49,-180.41C623.03,-175.45 258.75,-175.3 140.77,-179.97\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"140.61,-176.48 130.77,-180.41 140.91,-183.47 140.61,-176.48\"/>\n",
"<text text-anchor=\"middle\" x=\"418.63\" y=\"-168.99\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->12 -->\n",
"<g id=\"edge389\" class=\"edge\">\n",
"<title>1->12</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.3,-184.81C615.98,-196.95 209.11,-261.23 83.68,-285.98\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"82.9,-282.56 73.79,-287.97 84.28,-289.43 82.9,-282.56\"/>\n",
"<text text-anchor=\"middle\" x=\"389.99\" y=\"-224.2\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->11 -->\n",
"<g id=\"edge390\" class=\"edge\">\n",
"<title>1->11</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.45,-189.26C614.14,-219.97 192.04,-356.95 64.02,-403.65\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"62.45,-400.5 54.28,-407.24 64.87,-407.07 62.45,-400.5\"/>\n",
"<text text-anchor=\"middle\" x=\"387.48\" y=\"-285.25\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->10 -->\n",
"<g id=\"edge391\" class=\"edge\">\n",
"<title>1->10</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.33,-194.1C615.73,-244.16 206.77,-452.35 82.59,-521.04\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"80.85,-518.01 73.81,-525.93 84.25,-524.12 80.85,-518.01\"/>\n",
"<text text-anchor=\"middle\" x=\"389.46\" y=\"-346.37\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->9 -->\n",
"<g id=\"edge392\" class=\"edge\">\n",
"<title>1->9</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M725.44,-199.52C620.2,-269.62 239.57,-546.07 132.04,-630.37\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"129.59,-627.84 123.91,-636.78 133.93,-633.33 129.59,-627.84\"/>\n",
"<text text-anchor=\"middle\" x=\"443.24\" y=\"-403.74\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->8 -->\n",
"<g id=\"edge393\" class=\"edge\">\n",
"<title>1->8</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M730.69,-201.35C641.05,-283.47 302.81,-621.61 213.83,-717.9\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"211.1,-715.7 206.93,-725.44 216.26,-720.43 211.1,-715.7\"/>\n",
"<text text-anchor=\"middle\" x=\"457.76\" y=\"-448.43\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->7 -->\n",
"<g id=\"edge394\" class=\"edge\">\n",
"<title>1->7</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M736.04,-201.64C665.28,-289.76 389.24,-669.52 319.91,-773.91\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"316.95,-772.04 314.39,-782.32 322.8,-775.88 316.95,-772.04\"/>\n",
"<text text-anchor=\"middle\" x=\"513.47\" y=\"-476.57\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->6 -->\n",
"<g id=\"edge395\" class=\"edge\">\n",
"<title>1->6</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M740.4,-201.59C688.53,-291.26 487.93,-684.7 438.82,-792.67\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"435.57,-791.37 434.67,-801.93 441.96,-794.23 435.57,-791.37\"/>\n",
"<text text-anchor=\"middle\" x=\"575.11\" y=\"-485.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->5 -->\n",
"<g id=\"edge396\" class=\"edge\">\n",
"<title>1->5</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M744.03,-201.35C709.95,-288.52 587.05,-666.28 557.95,-772.59\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"554.56,-771.72 555.36,-782.28 561.32,-773.53 554.56,-771.72\"/>\n",
"<text text-anchor=\"middle\" x=\"636.49\" y=\"-475.77\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->4 -->\n",
"<g id=\"edge397\" class=\"edge\">\n",
"<title>1->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M746.99,-201.56C728.69,-283.16 675.91,-615.28 665.13,-715.06\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"661.62,-714.98 664.1,-725.28 668.58,-715.68 661.62,-714.98\"/>\n",
"<text text-anchor=\"middle\" x=\"691.56\" y=\"-447.11\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->3 -->\n",
"<g id=\"edge398\" class=\"edge\">\n",
"<title>1->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M749.73,-201.52C744.69,-273.57 744.49,-539.22 749.13,-626.69\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"745.63,-626.9 749.73,-636.67 752.62,-626.48 745.63,-626.9\"/>\n",
"<text text-anchor=\"middle\" x=\"734.93\" y=\"-402.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->2 -->\n",
"<g id=\"edge399\" class=\"edge\">\n",
"<title>1->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M752.24,-201.41C756.87,-259.81 785.74,-443.67 801.09,-514.66\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"797.73,-515.68 803.34,-524.67 804.56,-514.14 797.73,-515.68\"/>\n",
"<text text-anchor=\"middle\" x=\"762.16\" y=\"-346.83\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->1 -->\n",
"<g id=\"edge400\" class=\"edge\">\n",
"<title>1->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M779.78,-190.07C789.57,-190.22 797.54,-187.97 797.54,-183.33 797.54,-180.5 794.58,-178.56 790.05,-177.51\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"790.06,-174 779.78,-176.58 789.43,-180.97 790.06,-174\"/>\n",
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"</g>\n",
"</g>\n",
"</svg>"
],
"text/plain": [
"<Dot visualization: expr_as_graph [(\"K20\",(1..20)*(1..20))]>"
]
},
"execution_count": 76,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":dot expr_as_graph (\"K20\",(1..20)×(1..20))"
]
},
{
"cell_type": "code",
"execution_count": 77,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Preference changed: DOT_ENGINE = dot\n"
]
},
"execution_count": 77,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":pref DOT_ENGINE=dot"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Definitions- und Wertebereich\n",
"\n",
"Definitionsbereich (Domain in Englisch):\n",
"* $dom(r) = \\{a \\mid \\exists b.((a,b)\\in r)\\}$\n",
"\n",
"Wertebereich (Bildmenge, Range in Englisch):\n",
"* $ran(r) = \\{b \\mid \\exists a.((a,b)\\in r)\\}$\n",
" "
]
},
{
"cell_type": "code",
"execution_count": 78,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(2\\mapsto 1),(4\\mapsto 2),(6\\mapsto 3),(8\\mapsto 4),(10\\mapsto 5)\\}$"
],
"text/plain": [
"{(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)}"
]
},
"execution_count": 78,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":let h {a,b| a∈1..10 ∧ b∈1..10 & b*2=a}"
]
},
{
"cell_type": "code",
"execution_count": 79,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{2,4,6,8,10\\}$"
],
"text/plain": [
"{2,4,6,8,10}"
]
},
"execution_count": 79,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{a| ∃b.((a,b)∈h)}"
]
},
{
"cell_type": "code",
"execution_count": 80,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,2,3,4,5\\}$"
],
"text/plain": [
"{1,2,3,4,5}"
]
},
"execution_count": 80,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{b| ∃a.((a,b)∈h)}"
]
},
{
"cell_type": "code",
"execution_count": 81,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{2,4,6,8,10\\}$"
],
"text/plain": [
"{2,4,6,8,10}"
]
},
"execution_count": 81,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dom(h)"
]
},
{
"cell_type": "code",
"execution_count": 82,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(2\\mapsto 1),(4\\mapsto 2),(6\\mapsto 3),(8\\mapsto 4),(10\\mapsto 5)\\}$"
],
"text/plain": [
"{(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)}"
]
},
"execution_count": 82,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":let h {a,b| a∈1..10 ∧ b∈1..10 & b*2=a}"
]
},
{
"cell_type": "code",
"execution_count": 83,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
"<!-- Generated by graphviz version 2.44.1 (0)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
"<svg width=\"206pt\" height=\"305pt\"\n",
" viewBox=\"0.00 0.00 206.00 305.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 301)\">\n",
"<title>state</title>\n",
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"<!-- 10 -->\n",
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"<text text-anchor=\"middle\" x=\"27\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">10</text>\n",
"</g>\n",
"<!-- 5 -->\n",
"<g id=\"node2\" class=\"node\">\n",
"<title>5</title>\n",
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"<text text-anchor=\"middle\" x=\"27\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">5</text>\n",
"</g>\n",
"<!-- 10->5 -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>10->5</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-220.18 27,-210.18 23.5,-220.18 30.5,-220.18\"/>\n",
"<text text-anchor=\"middle\" x=\"32\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 8 -->\n",
"<g id=\"node3\" class=\"node\">\n",
"<title>8</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"126,-297 72,-297 72,-261 126,-261 126,-297\"/>\n",
"<text text-anchor=\"middle\" x=\"99\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">8</text>\n",
"</g>\n",
"<!-- 4 -->\n",
"<g id=\"node4\" class=\"node\">\n",
"<title>4</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"126,-210 72,-210 72,-174 126,-174 126,-210\"/>\n",
"<text text-anchor=\"middle\" x=\"99\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">4</text>\n",
"</g>\n",
"<!-- 8->4 -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>8->4</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-260.8C99,-249.16 99,-233.55 99,-220.24\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-220.18 99,-210.18 95.5,-220.18 102.5,-220.18\"/>\n",
"<text text-anchor=\"middle\" x=\"104\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 2 -->\n",
"<g id=\"node7\" class=\"node\">\n",
"<title>2</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"126,-123 72,-123 72,-87 126,-87 126,-123\"/>\n",
"<text text-anchor=\"middle\" x=\"99\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
"</g>\n",
"<!-- 4->2 -->\n",
"<g id=\"edge4\" class=\"edge\">\n",
"<title>4->2</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-133.18 99,-123.18 95.5,-133.18 102.5,-133.18\"/>\n",
"<text text-anchor=\"middle\" x=\"104\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 6 -->\n",
"<g id=\"node5\" class=\"node\">\n",
"<title>6</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"198,-297 144,-297 144,-261 198,-261 198,-297\"/>\n",
"<text text-anchor=\"middle\" x=\"171\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">6</text>\n",
"</g>\n",
"<!-- 3 -->\n",
"<g id=\"node6\" class=\"node\">\n",
"<title>3</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"198,-210 144,-210 144,-174 198,-174 198,-210\"/>\n",
"<text text-anchor=\"middle\" x=\"171\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
"</g>\n",
"<!-- 6->3 -->\n",
"<g id=\"edge3\" class=\"edge\">\n",
"<title>6->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M171,-260.8C171,-249.16 171,-233.55 171,-220.24\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"174.5,-220.18 171,-210.18 167.5,-220.18 174.5,-220.18\"/>\n",
"<text text-anchor=\"middle\" x=\"176\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g id=\"node8\" class=\"node\">\n",
"<title>1</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"126,-36 72,-36 72,0 126,0 126,-36\"/>\n",
"<text text-anchor=\"middle\" x=\"99\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
"</g>\n",
"<!-- 2->1 -->\n",
"<g id=\"edge5\" class=\"edge\">\n",
"<title>2->1</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-86.8C99,-75.16 99,-59.55 99,-46.24\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-46.18 99,-36.18 95.5,-46.18 102.5,-46.18\"/>\n",
"<text text-anchor=\"middle\" x=\"104\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"</g>\n",
"</svg>"
],
"text/plain": [
"<Dot visualization: expr_as_graph [hh={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"h\",h)]>"
]
},
"execution_count": 83,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":dot expr_as_graph (\"h\",h)"
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto\\text{\"nein\"}),(2\\mapsto\\text{\"nein\"}),(3\\mapsto\\text{\"ja\"}),(4\\mapsto\\text{\"nein\"}),(5\\mapsto\\text{\"nein\"}),(6\\mapsto\\text{\"ja\"}),(7\\mapsto\\text{\"nein\"})\\}$"
],
"text/plain": [
"{(1↦\"nein\"),(2↦\"nein\"),(3↦\"ja\"),(4↦\"nein\"),(5↦\"nein\"),(6↦\"ja\"),(7↦\"nein\")}"
]
},
"execution_count": 84,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":let d {x,y| x:1..7 ∧ (x mod 3 =0 => y = \"ja\") ∧ (x mod 3 >0 => y= \"nein\")}"
]
},
{
"cell_type": "code",
"execution_count": 85,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,2,3,4,5,6,7\\}$"
],
"text/plain": [
"{1,2,3,4,5,6,7}"
]
},
"execution_count": 85,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dom(d)"
]
},
{
"cell_type": "code",
"execution_count": 86,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{\\text{\"ja\"},\\text{\"nein\"}\\}$"
],
"text/plain": [
"{\"ja\",\"nein\"}"
]
},
"execution_count": 86,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ran(d)"
]
},
{
"cell_type": "code",
"execution_count": 87,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
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"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
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"<!-- Generated by graphviz version 2.44.1 (0)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
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" viewBox=\"0.00 0.00 494.00 131.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
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"<title>state</title>\n",
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"<title>7->"nein"</title>\n",
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"<title>6->"ja"</title>\n",
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"<title>5->"nein"</title>\n",
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"</g>\n",
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"</g>\n",
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"<title>3->"ja"</title>\n",
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"<title>2->"nein"</title>\n",
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"<title>1->"nein"</title>\n",
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"</g>\n",
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"text/plain": [
"<Dot visualization: expr_as_graph [dhd={(1,\"nein\"),(2,\"nein\"),(3,\"ja\"),(4,\"nein\"),(5,\"nein\"),(6,\"ja\"),(7,\"nein\")} & h={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"d\",d)]>"
]
},
"execution_count": 87,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":dot expr_as_graph (\"d\",d)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Relationales Abbild und Umkehrrelation\n",
" \n",
"Abbild:\n",
" * $r[A] = \\{b \\mid \\exists a.((a,b)\\in r \\wedge a\\in A)\\}$\n",
"\n",
"Umkehrrelation:\n",
" * $r^{-1} = \\{(b,a) \\mid (a,b)\\in r \\}$\n",
"\n",
"Im Notebook muss für die Umkehrrelation leider Tilde (~) verwendet werden: "
]
},
{
"cell_type": "code",
"execution_count": 88,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{4,5\\}$"
],
"text/plain": [
"{4,5}"
]
},
"execution_count": 88,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"h[8..10]"
]
},
{
"cell_type": "code",
"execution_count": 89,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{\\text{\"ja\"}\\}$"
],
"text/plain": [
"{\"ja\"}"
]
},
"execution_count": 89,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"d[{3,6}]"
]
},
{
"cell_type": "code",
"execution_count": 90,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$"
],
"text/plain": [
"∅"
]
},
"execution_count": 90,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"d[{0}]"
]
},
{
"cell_type": "code",
"execution_count": 91,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(\\text{\"ja\"}\\mapsto 3),(\\text{\"ja\"}\\mapsto 6),(\\text{\"nein\"}\\mapsto 1),(\\text{\"nein\"}\\mapsto 2),(\\text{\"nein\"}\\mapsto 4),(\\text{\"nein\"}\\mapsto 5),(\\text{\"nein\"}\\mapsto 7)\\}$"
],
"text/plain": [
"{(\"ja\"↦3),(\"ja\"↦6),(\"nein\"↦1),(\"nein\"↦2),(\"nein\"↦4),(\"nein\"↦5),(\"nein\"↦7)}"
]
},
"execution_count": 91,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"d~"
]
},
{
"cell_type": "code",
"execution_count": 92,
"metadata": {},
"outputs": [
{
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"<Dot visualization: expr_as_graph [dhd={(1,\"nein\"),(2,\"nein\"),(3,\"ja\"),(4,\"nein\"),(5,\"nein\"),(6,\"ja\"),(7,\"nein\")} & h={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"d~\",d~)]>"
]
},
"execution_count": 92,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":dot expr_as_graph (\"d~\",d~)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Man kann natürlich auch beide Operatoren verknüpfen, zum Beispiel um herauszufinden welche Zahlen durch 3 teilbar sind:"
]
},
{
"cell_type": "code",
"execution_count": 93,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{3,6\\}$"
],
"text/plain": [
"{3,6}"
]
},
"execution_count": 93,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"d~[{\"ja\"}]"
]
},
{
"cell_type": "code",
"execution_count": 94,
"metadata": {},
"outputs": [],
"source": [
":unlet d"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Binäre Relationen - Verknüpfung\n",
" \n",
"Wir können Relationen $r_1$ und $r_2$ mit dem Operator ``;'' verknüpfen:\n",
"* $(r_1 ; r_2)$ = $\\{(a,c) \\mid \\exists b. ( (a,b)\\in r_1 \\wedge (b,c)\\in r_2)\\}$.\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 95,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto 4),(1\\mapsto 8)\\}$"
],
"text/plain": [
"{(1↦4),(1↦8)}"
]
},
"execution_count": 95,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"({1↦2, 2↦3} ; {2↦4, 2↦8})"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"Man kann eine Relation auch mit sich selber verknüpfen wenn Werte- und Definitionsbereich kompatibel sind.\n",
"Zum Beispiel $h^2 = (h;h)$:"
]
},
{
"cell_type": "code",
"execution_count": 96,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(4\\mapsto 1),(8\\mapsto 2)\\}$"
],
"text/plain": [
"{(4↦1),(8↦2)}"
]
},
"execution_count": 96,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(h ; h)"
]
},
{
"cell_type": "code",
"execution_count": 97,
"metadata": {},
"outputs": [
{
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"source": [
":dot expr_as_graph (\"h\",h,\"hh\",(h;h))"
]
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{
"cell_type": "code",
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"cell_type": "markdown",
"metadata": {},
"source": [
"# Transitive Hülle\n",
" \n",
"Gegeben eine Relation $r$ von $A$ nach $A$\n",
"* $r^1 = r$\n",
"* $r^k = (r^{k-1} ; r) = (r ; r^{k-1})$\n",
"\n",
"* $r^+$ = $\\bigcup_{i\\geq 1} r^i$"
]
},
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"metadata": {},
"outputs": [
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],
"text/plain": [
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]
},
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"metadata": {},
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"source": [
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]
},
{
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"metadata": {},
"outputs": [
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],
"text/plain": [
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]
},
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]
},
{
"cell_type": "code",
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"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$"
],
"text/plain": [
"∅"
]
},
"execution_count": 102,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(h;h;h;h)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Die transitive Hülle wird in B als ```closure1``` geschrieben:"
]
},
{
"cell_type": "code",
"execution_count": 103,
"metadata": {},
"outputs": [
{
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"$\\{(2\\mapsto 1),(4\\mapsto 1),(4\\mapsto 2),(6\\mapsto 3),(8\\mapsto 1),(8\\mapsto 2),(8\\mapsto 4),(10\\mapsto 5)\\}$"
],
"text/plain": [
"{(2↦1),(4↦1),(4↦2),(6↦3),(8↦1),(8↦2),(8↦4),(10↦5)}"
]
},
"execution_count": 103,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"closure1(h)"
]
},
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"cell_type": "code",
"execution_count": 104,
"metadata": {},
"outputs": [
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"<title>8->4</title>\n",
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"text/plain": [
"<Dot visualization: expr_as_graph [hh={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"h+\",closure1(h))]>"
]
},
"execution_count": 104,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":dot expr_as_graph (\"h+\",closure1(h))"
]
},
{
"cell_type": "code",
"execution_count": 105,
"metadata": {},
"outputs": [],
"source": [
":unlet h"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Ein weiteres Beispiel ist folgende abgeänderte strikte Untermengenrelation $\\subset$ für $ℙ(1 .. 3) $:"
]
},
{
"cell_type": "code",
"execution_count": 106,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\{(\\emptyset\\mapsto\\{1\\}),(\\emptyset\\mapsto\\{2\\}),(\\emptyset\\mapsto\\{3\\}),(\\{1\\}\\mapsto\\{1,2\\}),(\\{1\\}\\mapsto\\{1,3\\}),(\\{1,2\\}\\mapsto\\{1,2,3\\}),(\\{1,3\\}\\mapsto\\{1,2,3\\}),(\\{2\\}\\mapsto\\{1,2\\}),(\\{2\\}\\mapsto\\{2,3\\}),(\\{2,3\\}\\mapsto\\{1,2,3\\}),(\\{3\\}\\mapsto\\{1,3\\}),(\\{3\\}\\mapsto\\{2,3\\})\\}$"
],
"text/plain": [
"{(∅↦{1}),(∅↦{2}),(∅↦{3}),({1}↦{1,2}),({1}↦{1,3}),({1,2}↦{1,2,3}),({1,3}↦{1,2,3}),({2}↦{1,2}),({2}↦{2,3}),({2,3}↦{1,2,3}),({3}↦{1,3}),({3}↦{2,3})}"
]
},
"execution_count": 106,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":let sub1 {x,y|y:ℙ(1..3) & x ⊂ y & card(x)+1=card(y)}"
]
},
{
"cell_type": "code",
"execution_count": 107,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Preference changed: DOT_DECOMPOSE_NODES = false\n"
]
},
"execution_count": 107,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":pref DOT_DECOMPOSE_NODES=false"
]
},
{
"cell_type": "code",
"execution_count": 108,
"metadata": {},
"outputs": [
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],
"text/plain": [
"<Dot visualization: expr_as_graph [sub1sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})}(\"sub1\",sub1)]>"
]
},
"execution_count": 108,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":dot expr_as_graph (\"sub1\",sub1)"
]
},
{
"cell_type": "code",
"execution_count": 109,
"metadata": {},
"outputs": [
{
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"<!-- \\{3\\} -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>\\{3\\}</title>\n",
"<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"210,-210 156,-210 156,-174 210,-174 210,-210\"/>\n",
"<text text-anchor=\"middle\" x=\"183\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">{3}</text>\n",
"</g>\n",
"<!-- \\{2,3\\} -->\n",
"<g id=\"node2\" class=\"node\">\n",
"<title>\\{2,3\\}</title>\n",
"<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"127.5,-123 70.5,-123 70.5,-87 127.5,-87 127.5,-123\"/>\n",
"<text text-anchor=\"middle\" x=\"99\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">{2,3}</text>\n",
"</g>\n",
"<!-- \\{3\\}->\\{2,3\\} -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>\\{3\\}->\\{2,3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M155.73,-179.03C144.85,-173.26 132.84,-165.5 124,-156 117.73,-149.26 112.67,-140.65 108.79,-132.48\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"111.97,-131.01 104.77,-123.23 105.55,-133.79 111.97,-131.01\"/>\n",
"<text text-anchor=\"middle\" x=\"141.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{1,2,3\\} -->\n",
"<g id=\"node3\" class=\"node\">\n",
"<title>\\{1,2,3\\}</title>\n",
"<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"291.5,-36 220.5,-36 220.5,0 291.5,0 291.5,-36\"/>\n",
"<text text-anchor=\"middle\" x=\"256\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1,2,3}</text>\n",
"</g>\n",
"<!-- \\{3\\}->\\{1,2,3\\} -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>\\{3\\}->\\{1,2,3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M179,-173.95C174.87,-152.63 170.61,-115.48 183,-87 190.74,-69.21 205.68,-53.97 220.04,-42.45\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"222.54,-44.95 228.37,-36.11 218.3,-39.38 222.54,-44.95\"/>\n",
"<text text-anchor=\"middle\" x=\"200.5\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{1,3\\} -->\n",
"<g id=\"node4\" class=\"node\">\n",
"<title>\\{1,3\\}</title>\n",
"<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"404.5,-123 347.5,-123 347.5,-87 404.5,-87 404.5,-123\"/>\n",
"<text text-anchor=\"middle\" x=\"376\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1,3}</text>\n",
"</g>\n",
"<!-- \\{3\\}->\\{1,3\\} -->\n",
"<g id=\"edge3\" class=\"edge\">\n",
"<title>\\{3\\}->\\{1,3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M210.07,-181.27C217.75,-178.7 226.15,-176.08 234,-174 273.86,-163.46 288.53,-175.24 325,-156 331.55,-152.55 343.09,-141.43 353.54,-130.56\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"356.2,-132.84 360.52,-123.16 351.11,-128.03 356.2,-132.84\"/>\n",
"<text text-anchor=\"middle\" x=\"360.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{2,3\\}->\\{1,2,3\\} -->\n",
"<g id=\"edge4\" class=\"edge\">\n",
"<title>\\{2,3\\}->\\{1,2,3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M103.58,-86.83C107.35,-75.92 113.76,-62.33 124,-54 148.29,-34.23 182.67,-25.54 210.25,-21.76\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"210.94,-25.21 220.46,-20.54 210.11,-18.26 210.94,-25.21\"/>\n",
"<text text-anchor=\"middle\" x=\"141.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{1,3\\}->\\{1,2,3\\} -->\n",
"<g id=\"edge8\" class=\"edge\">\n",
"<title>\\{1,3\\}->\\{1,2,3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M351.72,-86.8C333.5,-73.9 308.38,-56.1 288.44,-41.98\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"290.43,-39.1 280.25,-36.18 286.38,-44.81 290.43,-39.1\"/>\n",
"<text text-anchor=\"middle\" x=\"341.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{2\\} -->\n",
"<g id=\"node5\" class=\"node\">\n",
"<title>\\{2\\}</title>\n",
"<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"54,-210 0,-210 0,-174 54,-174 54,-210\"/>\n",
"<text text-anchor=\"middle\" x=\"27\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">{2}</text>\n",
"</g>\n",
"<!-- \\{2\\}->\\{2,3\\} -->\n",
"<g id=\"edge5\" class=\"edge\">\n",
"<title>\\{2\\}->\\{2,3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M28.6,-173.84C30.3,-163.47 33.79,-150.47 41,-141 46.43,-133.87 53.76,-127.88 61.38,-122.99\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"63.5,-125.81 70.37,-117.75 59.97,-119.76 63.5,-125.81\"/>\n",
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"</g>\n",
"<!-- \\{2\\}->\\{1,2,3\\} -->\n",
"<g id=\"edge6\" class=\"edge\">\n",
"<title>\\{2\\}->\\{1,2,3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M21.33,-173.83C15.16,-151.7 8.36,-113 26,-87 46.72,-56.46 149,-35.62 209.97,-25.68\"/>\n",
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"</g>\n",
"<!-- \\{1,2\\} -->\n",
"<g id=\"node6\" class=\"node\">\n",
"<title>\\{1,2\\}</title>\n",
"<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"284.5,-123 227.5,-123 227.5,-87 284.5,-87 284.5,-123\"/>\n",
"<text text-anchor=\"middle\" x=\"256\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1,2}</text>\n",
"</g>\n",
"<!-- \\{2\\}->\\{1,2\\} -->\n",
"<g id=\"edge7\" class=\"edge\">\n",
"<title>\\{2\\}->\\{1,2\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M54.06,-178.86C58.66,-177.07 63.42,-175.37 68,-174 109.18,-161.71 122.49,-170.34 163,-156 182.83,-148.98 203.57,-138.21 220.42,-128.46\"/>\n",
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"</g>\n",
"<!-- \\{1,2\\}->\\{1,2,3\\} -->\n",
"<g id=\"edge9\" class=\"edge\">\n",
"<title>\\{1,2\\}->\\{1,2,3\\}</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"259.5,-46.18 256,-36.18 252.5,-46.18 259.5,-46.18\"/>\n",
"<text text-anchor=\"middle\" x=\"273.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{1\\} -->\n",
"<g id=\"node7\" class=\"node\">\n",
"<title>\\{1\\}</title>\n",
"<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"393,-210 339,-210 339,-174 393,-174 393,-210\"/>\n",
"<text text-anchor=\"middle\" x=\"366\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1}</text>\n",
"</g>\n",
"<!-- \\{1\\}->\\{1,2,3\\} -->\n",
"<g id=\"edge10\" class=\"edge\">\n",
"<title>\\{1\\}->\\{1,2,3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M393.19,-181.3C404.53,-175.76 416.54,-167.55 423,-156 431.54,-140.74 432.06,-114.52 414,-87 389.36,-49.46 339.16,-32.56 301.92,-25\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"302.27,-21.51 291.79,-23.11 300.98,-28.39 302.27,-21.51\"/>\n",
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"</g>\n",
"<!-- \\{1\\}->\\{1,3\\} -->\n",
"<g id=\"edge11\" class=\"edge\">\n",
"<title>\\{1\\}->\\{1,3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M376.28,-173.77C379.02,-168.28 381.61,-162.06 383,-156 384.68,-148.68 384.46,-140.67 383.41,-133.24\"/>\n",
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"<text text-anchor=\"middle\" x=\"401.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{1\\}->\\{1,2\\} -->\n",
"<g id=\"edge12\" class=\"edge\">\n",
"<title>\\{1\\}->\\{1,2\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M338.64,-182.51C323.19,-176.77 304.19,-168.02 290,-156 282.03,-149.25 275.12,-140.2 269.65,-131.66\"/>\n",
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"<text text-anchor=\"middle\" x=\"307.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\} -->\n",
"<g id=\"node8\" class=\"node\">\n",
"<title>\\{\\}</title>\n",
"<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"283,-297 229,-297 229,-261 283,-261 283,-297\"/>\n",
"<text text-anchor=\"middle\" x=\"256\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">{}</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{3\\} -->\n",
"<g id=\"edge13\" class=\"edge\">\n",
"<title>\\{\\}->\\{3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M228.8,-262.2C221.08,-256.81 213.12,-250.28 207,-243 201.27,-236.19 196.55,-227.78 192.86,-219.81\"/>\n",
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"</g>\n",
"<!-- \\{\\}->\\{2,3\\} -->\n",
"<g id=\"edge14\" class=\"edge\">\n",
"<title>\\{\\}->\\{2,3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M228.96,-273.34C195.64,-266.07 139.98,-248.62 112,-210 95.96,-187.87 94.27,-155.9 95.61,-133.26\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"99.1,-133.54 96.43,-123.28 92.12,-132.96 99.1,-133.54\"/>\n",
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"</g>\n",
"<!-- \\{\\}->\\{1,2,3\\} -->\n",
"<g id=\"edge15\" class=\"edge\">\n",
"<title>\\{\\}->\\{1,2,3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M283.15,-278.74C329.94,-278.34 424.61,-269.75 465,-210 495.62,-164.7 495.45,-133.69 467,-87 449.6,-58.46 358.68,-37.34 302.02,-26.72\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"302.33,-23.22 291.86,-24.86 301.07,-30.1 302.33,-23.22\"/>\n",
"<text text-anchor=\"middle\" x=\"505.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{1,3\\} -->\n",
"<g id=\"edge17\" class=\"edge\">\n",
"<title>\\{\\}->\\{1,3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M283.09,-273.23C316.56,-265.84 372.64,-248.27 402,-210 421.51,-184.57 437.38,-169.65 423,-141 420.49,-136 416.89,-131.53 412.78,-127.6\"/>\n",
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"</g>\n",
"<!-- \\{\\}->\\{2\\} -->\n",
"<g id=\"edge16\" class=\"edge\">\n",
"<title>\\{\\}->\\{2\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M228.69,-277.12C177.15,-274.9 68.28,-267.39 41,-243 34.5,-237.19 30.92,-228.64 28.97,-220.24\"/>\n",
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"</g>\n",
"<!-- \\{\\}->\\{1,2\\} -->\n",
"<g id=\"edge18\" class=\"edge\">\n",
"<title>\\{\\}->\\{1,2\\}</title>\n",
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"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"259.5,-133.05 256,-123.05 252.5,-133.05 259.5,-133.05\"/>\n",
"<text text-anchor=\"middle\" x=\"273.5\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{1\\} -->\n",
"<g id=\"edge19\" class=\"edge\">\n",
"<title>\\{\\}->\\{1\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M278.26,-260.8C294.81,-248.01 317.58,-230.42 335.77,-216.36\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"338,-219.06 343.77,-210.18 333.72,-213.52 338,-219.06\"/>\n",
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"</g>\n",
"</g>\n",
"</svg>"
],
"text/plain": [
"<Dot visualization: expr_as_graph [sub1sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})}(\"sub1\",closure1(sub1))]>"
]
},
"execution_count": 109,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":dot expr_as_graph (\"sub1\",closure1(sub1))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Transitive und Reflexive Hülle\n",
" \n",
"Gegeben eine Relation $r$ von $A$ nach $A$\n",
"* $r^0 = \\{(a,a) \\mid a\\in A\\}$\n",
"* $r^1 = r$\n",
"* $r^k = (r^{k-1} ; r) = (r ; r^{k-1})$\n",
"* $r^*$ = $\\bigcup_{i\\geq 0} r^i$\n"
]
},
{
"cell_type": "code",
"execution_count": 110,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,2,3\\}$"
],
"text/plain": [
"{1,2,3}"
]
},
"execution_count": 110,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":let A {1,2,3}"
]
},
{
"cell_type": "code",
"execution_count": 111,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto 2),(2\\mapsto 3)\\}$"
],
"text/plain": [
"{(1↦2),(2↦3)}"
]
},
"execution_count": 111,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":let r {(1,2), (2,3)}"
]
},
{
"cell_type": "code",
"execution_count": 112,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
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" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
"<!-- Generated by graphviz version 2.44.1 (0)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
"<svg width=\"62pt\" height=\"218pt\"\n",
" viewBox=\"0.00 0.00 62.00 218.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 214)\">\n",
"<title>state</title>\n",
"<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-214 58,-214 58,4 -4,4\"/>\n",
"<!-- 2 -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>2</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-123 0,-123 0,-87 54,-87 54,-123\"/>\n",
"<text text-anchor=\"middle\" x=\"27\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
"</g>\n",
"<!-- 3 -->\n",
"<g id=\"node2\" class=\"node\">\n",
"<title>3</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-36 0,-36 0,0 54,0 54,-36\"/>\n",
"<text text-anchor=\"middle\" x=\"27\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
"</g>\n",
"<!-- 2->3 -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>2->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-86.8C27,-75.16 27,-59.55 27,-46.24\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-46.18 27,-36.18 23.5,-46.18 30.5,-46.18\"/>\n",
"<text text-anchor=\"middle\" x=\"30.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">r</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g id=\"node3\" class=\"node\">\n",
"<title>1</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-210 0,-210 0,-174 54,-174 54,-210\"/>\n",
"<text text-anchor=\"middle\" x=\"27\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
"</g>\n",
"<!-- 1->2 -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>1->2</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-173.8C27,-162.16 27,-146.55 27,-133.24\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-133.18 27,-123.18 23.5,-133.18 30.5,-133.18\"/>\n",
"<text text-anchor=\"middle\" x=\"30.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">r</text>\n",
"</g>\n",
"</g>\n",
"</svg>"
],
"text/plain": [
"<Dot visualization: expr_as_graph [Arsub1A={1,2,3} & r={(1,2),(2,3)} & sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})}(\"r\",r)]>"
]
},
"execution_count": 112,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":dot expr_as_graph (\"r\",r)"
]
},
{
"cell_type": "code",
"execution_count": 113,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto 3)\\}$"
],
"text/plain": [
"{(1↦3)}"
]
},
"execution_count": 113,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(r;r)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In B kann man $r^0$ auch als die Identitätsrelation über A beschreiben:"
]
},
{
"cell_type": "code",
"execution_count": 114,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto 1),(2\\mapsto 2),(3\\mapsto 3)\\}$"
],
"text/plain": [
"{(1↦1),(2↦2),(3↦3)}"
]
},
"execution_count": 114,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"id(A)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Die transitive und reflexive Hülle ist hier:"
]
},
{
"cell_type": "code",
"execution_count": 115,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto 1),(1\\mapsto 2),(1\\mapsto 3),(2\\mapsto 2),(2\\mapsto 3),(3\\mapsto 3)\\}$"
],
"text/plain": [
"{(1↦1),(1↦2),(1↦3),(2↦2),(2↦3),(3↦3)}"
]
},
"execution_count": 115,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"id(A) \\/ r \\/ (r;r) \\/ (r;r;r) \\/ (r;r;r;r)"
]
},
{
"cell_type": "code",
"execution_count": 116,
"metadata": {},
"outputs": [
{
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]
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"execution_count": 116,
"metadata": {},
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}
],
"source": [
":dot expr_as_graph (\"r*\",id(A) \\/ r \\/ (r;r) \\/ (r;r;r) \\/ (r;r;r;r))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Als Vergleich dazu sieht die transitive Hülle so aus:"
]
},
{
"cell_type": "code",
"execution_count": 117,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto 2),(1\\mapsto 3),(2\\mapsto 3)\\}$"
],
"text/plain": [
"{(1↦2),(1↦3),(2↦3)}"
]
},
"execution_count": 117,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"closure1(r)"
]
},
{
"cell_type": "code",
"execution_count": 118,
"metadata": {},
"outputs": [
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"execution_count": 118,
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],
"source": [
":dot expr_as_graph (\"r+\",closure1(r))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Man kann die transitive Hülle natürlich mit den anderen Operatoren verknüpfen, zum Beispiel um auszurechnen von welchen Knoten aus man den Knoten 3 erreichen kann:"
]
},
{
"cell_type": "code",
"execution_count": 119,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,2\\}$"
],
"text/plain": [
"{1,2}"
]
},
"execution_count": 119,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"closure1(r)~[{3}]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Anmerkung: die reflexive und transitive Hülle in B wird als ```closure``` geschrieben, ist bei Relationen über Zahlen immer unendlich."
]
},
{
"cell_type": "code",
"execution_count": 120,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$/*@symbolic*/ \\{\\mathit{z\\_},\\mathit{z\\_\\_}\\mid \\mathit{z\\_} \\mapsto \\mathit{z\\_\\_} \\in \\{(1\\mapsto 2),(1\\mapsto 3),(2\\mapsto 3)\\} \\lor \\mathit{z\\_} = \\mathit{z\\_\\_}\\}$"
],
"text/plain": [
"/*@symbolic*/ {z_,z__∣z_ ↦ z__ ∈ {(1↦2),(1↦3),(2↦3)} ∨ z_ = z__}"
]
},
"execution_count": 120,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"closure(r)"
]
},
{
"cell_type": "code",
"execution_count": 121,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,2,3\\}$"
],
"text/plain": [
"{1,2,3}"
]
},
"execution_count": 121,
"metadata": {},
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}
],
"source": [
"closure(r)~[{3}]"
]
},
{
"cell_type": "code",
"execution_count": 122,
"metadata": {},
"outputs": [],
"source": [
":unlet r"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Funktionen\n",
" \n",
"Was unterscheidet Funktionen von Relationen?\n",
"Wie kann man Funktionen in Mengentheorie und Logik darstellen?\n",
"\n",
"Eine Funktion kann als Menge an Paaren angesehen werden.\n",
"Zum Beispiel ist die Inkrementfunktion (+1) eine unendliche Mengen an Paaren.\n",
"Die linke Komponente des Paares ist die Eingabe der Funktion, die rechte Komponente die Ausgabe.\n",
"Eingeschränkt auf den Bereich 1..10 sieht dies Funktion so aus:"
]
},
{
"cell_type": "code",
"execution_count": 123,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto 2),(2\\mapsto 3),(3\\mapsto 4),(4\\mapsto 5),(5\\mapsto 6),(6\\mapsto 7),(7\\mapsto 8),(8\\mapsto 9),(9\\mapsto 10),(10\\mapsto 11)\\}$"
],
"text/plain": [
"{(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
"execution_count": 123,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{x,y|x:1..10 & y=x+1}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Eine Funktion kann also als Menge von Paaren dargestellt werden. Es sind aber nicht alle Mengen an Paaren auch Funktionen. Zum Beispiel, unsere Relation sub1 von oben ist keine Funktion:"
]
},
{
"cell_type": "code",
"execution_count": 124,
"metadata": {},
"outputs": [
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"<text text-anchor=\"middle\" x=\"222.17\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1,2}</text>\n",
"</g>\n",
"<!-- \\{2\\}->\\{1,2\\} -->\n",
"<g id=\"edge5\" class=\"edge\">\n",
"<title>\\{2\\}->\\{1,2\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M258.3,-173.8C268.19,-164.53 276.08,-152.63 270.17,-141 267.41,-135.58 263.38,-130.8 258.81,-126.66\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"260.75,-123.73 250.73,-120.27 256.41,-129.22 260.75,-123.73\"/>\n",
"<text text-anchor=\"middle\" x=\"289.67\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{1,2\\}->\\{1,2,3\\} -->\n",
"<g id=\"edge7\" class=\"edge\">\n",
"<title>\\{1,2\\}->\\{1,2,3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M211.03,-86.64C204.32,-76.69 195.32,-64.19 186.17,-54 182.85,-50.3 179.17,-46.59 175.44,-43.04\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"177.53,-40.2 167.79,-36.02 172.79,-45.36 177.53,-40.2\"/>\n",
"<text text-anchor=\"middle\" x=\"214.67\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{1\\} -->\n",
"<g id=\"node7\" class=\"node\">\n",
"<title>\\{1\\}</title>\n",
"<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"178.17,-210 124.17,-210 124.17,-174 178.17,-174 178.17,-210\"/>\n",
"<text text-anchor=\"middle\" x=\"151.17\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1}</text>\n",
"</g>\n",
"<!-- \\{1\\}->\\{1,3\\} -->\n",
"<g id=\"edge8\" class=\"edge\">\n",
"<title>\\{1\\}->\\{1,3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M135.18,-173.8C123.62,-161.36 107.83,-144.36 94.94,-130.5\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"97.5,-128.12 88.13,-123.18 92.38,-132.88 97.5,-128.12\"/>\n",
"<text text-anchor=\"middle\" x=\"133.67\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{1\\}->\\{1,2\\} -->\n",
"<g id=\"edge9\" class=\"edge\">\n",
"<title>\\{1\\}->\\{1,2\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M157.95,-173.69C162.45,-163.51 169.03,-150.77 177.17,-141 180.49,-137.02 184.34,-133.19 188.35,-129.64\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"190.72,-132.22 196.2,-123.15 186.26,-126.83 190.72,-132.22\"/>\n",
"<text text-anchor=\"middle\" x=\"194.67\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\} -->\n",
"<g id=\"node8\" class=\"node\">\n",
"<title>\\{\\}</title>\n",
"<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"178.17,-297 124.17,-297 124.17,-261 178.17,-261 178.17,-297\"/>\n",
"<text text-anchor=\"middle\" x=\"151.17\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">{}</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{3\\} -->\n",
"<g id=\"edge10\" class=\"edge\">\n",
"<title>\\{\\}->\\{3\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M131.61,-260.68C125.36,-255.09 118.44,-248.82 112.17,-243 103.32,-234.78 93.75,-225.65 85.27,-217.47\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"87.46,-214.72 77.83,-210.28 82.59,-219.75 87.46,-214.72\"/>\n",
"<text text-anchor=\"middle\" x=\"129.67\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{2\\} -->\n",
"<g id=\"edge11\" class=\"edge\">\n",
"<title>\\{\\}->\\{2\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M171.03,-260.99C177.3,-255.41 184.15,-249.08 190.17,-243 197.98,-235.11 206.12,-226.11 213.25,-217.94\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"216.11,-219.97 219.98,-210.11 210.8,-215.41 216.11,-219.97\"/>\n",
"<text text-anchor=\"middle\" x=\"220.67\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{1\\} -->\n",
"<g id=\"edge12\" class=\"edge\">\n",
"<title>\\{\\}->\\{1\\}</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M151.17,-260.8C151.17,-249.16 151.17,-233.55 151.17,-220.24\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"154.67,-220.18 151.17,-210.18 147.67,-220.18 154.67,-220.18\"/>\n",
"<text text-anchor=\"middle\" x=\"168.67\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"</g>\n",
"</svg>"
],
"text/plain": [
"<Dot visualization: expr_as_graph [Asub1A={1,2,3} & sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})}(\"sub1\",sub1)]>"
]
},
"execution_count": 124,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":dot expr_as_graph (\"sub1\",sub1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Die Eingabe ```{1}``` hat zwei mögliche Ausgaben (Nachfolger): ```{1,3}``` und ```{1,2}```. Die Eingabe $\\emptyset$ hat drei mögliche Ausgaben:"
]
},
{
"cell_type": "code",
"execution_count": 125,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\{(\\emptyset\\mapsto\\{1\\}),(\\emptyset\\mapsto\\{2\\}),(\\emptyset\\mapsto\\{3\\}),(\\{1\\}\\mapsto\\{1,2\\}),(\\{1\\}\\mapsto\\{1,3\\}),(\\{1,2\\}\\mapsto\\{1,2,3\\}),(\\{1,3\\}\\mapsto\\{1,2,3\\}),(\\{2\\}\\mapsto\\{1,2\\}),(\\{2\\}\\mapsto\\{2,3\\}),(\\{2,3\\}\\mapsto\\{1,2,3\\}),(\\{3\\}\\mapsto\\{1,3\\}),(\\{3\\}\\mapsto\\{2,3\\})\\}$"
],
"text/plain": [
"{(∅↦{1}),(∅↦{2}),(∅↦{3}),({1}↦{1,2}),({1}↦{1,3}),({1,2}↦{1,2,3}),({1,3}↦{1,2,3}),({2}↦{1,2}),({2}↦{2,3}),({2,3}↦{1,2,3}),({3}↦{1,3}),({3}↦{2,3})}"
]
},
"execution_count": 125,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sub1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Eine (totale) Funktion $F$ von $A$ nach $B$ ist \n",
"* eine Relation von $A$ nach $B$ (also eine Untermenge von $A\\times B$),so dass\n",
" * $\\forall a.( a\\in A \\Rightarrow \\exists b.( (a,b) \\in F))$\n",
" * $\\forall (a,b,c).(((a,b)\\in F \\wedge (a,c)\\in F) \\Rightarrow b=c)$\n",
" \n",
" Wir schreiben dann $F \\in A \\rightarrow B$."
]
},
{
"cell_type": "code",
"execution_count": 126,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{FALSE}$"
],
"text/plain": [
"FALSE"
]
},
"execution_count": 126,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"∀(a,b,c).((a,b)∈sub1 & (a,c)∈sub1 ⇒ b=c)"
]
},
{
"cell_type": "code",
"execution_count": 127,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{FALSE}$"
],
"text/plain": [
"FALSE"
]
},
"execution_count": 127,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sub1 ∈ ℙ(1..3) --> ℙ(1..3)"
]
},
{
"cell_type": "code",
"execution_count": 128,
"metadata": {},
"outputs": [],
"source": [
":unlet sub1"
]
},
{
"cell_type": "code",
"execution_count": 129,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto 2),(2\\mapsto 3),(3\\mapsto 4),(4\\mapsto 5),(5\\mapsto 6),(6\\mapsto 7),(7\\mapsto 8),(8\\mapsto 9),(9\\mapsto 10),(10\\mapsto 11)\\}$"
],
"text/plain": [
"{(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
"execution_count": 129,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":let i10 {x,y|x:1..10 & y=x+1}"
]
},
{
"cell_type": "code",
"execution_count": 130,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{A} = \\{1,2,3\\}$\n",
"* $\\mathit{i10} = \\{(1\\mapsto 2),(2\\mapsto 3),(3\\mapsto 4),(4\\mapsto 5),(5\\mapsto 6),(6\\mapsto 7),(7\\mapsto 8),(8\\mapsto 9),(9\\mapsto 10),(10\\mapsto 11)\\}$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
"\tA = {1,2,3}\n",
"\ti10 = {(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
"execution_count": 130,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"∀(a,b,c).((a,b)∈i10 & (a,c)∈i10 ⇒ b=c)"
]
},
{
"cell_type": "code",
"execution_count": 131,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{A} = \\{1,2,3\\}$\n",
"* $\\mathit{i10} = \\{(1\\mapsto 2),(2\\mapsto 3),(3\\mapsto 4),(4\\mapsto 5),(5\\mapsto 6),(6\\mapsto 7),(7\\mapsto 8),(8\\mapsto 9),(9\\mapsto 10),(10\\mapsto 11)\\}$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
"\tA = {1,2,3}\n",
"\ti10 = {(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
"execution_count": 131,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"i10 ∈ 1..10 --> 2..11"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Anmerkung: es wird nicht geprüft ob der komplette Wertebereich abgedeckt wird:"
]
},
{
"cell_type": "code",
"execution_count": 132,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{A} = \\{1,2,3\\}$\n",
"* $\\mathit{i10} = \\{(1\\mapsto 2),(2\\mapsto 3),(3\\mapsto 4),(4\\mapsto 5),(5\\mapsto 6),(6\\mapsto 7),(7\\mapsto 8),(8\\mapsto 9),(9\\mapsto 10),(10\\mapsto 11)\\}$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
"\tA = {1,2,3}\n",
"\ti10 = {(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
"execution_count": 132,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"i10 ∈ 1..10 --> 0..100"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Es wird aber wohl geprüft ob der Definitionsbereich komplett abgedeckt wird:"
]
},
{
"cell_type": "code",
"execution_count": 133,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{FALSE}$"
],
"text/plain": [
"FALSE"
]
},
"execution_count": 133,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"i10 ∈ 2..10 --> 0..100"
]
},
{
"cell_type": "code",
"execution_count": 134,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{FALSE}$"
],
"text/plain": [
"FALSE"
]
},
"execution_count": 134,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"i10 ∈ 0..10 --> 0..100"
]
},
{
"cell_type": "code",
"execution_count": 135,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,2,3,4,5,6,7,8,9,10\\}$"
],
"text/plain": [
"{1,2,3,4,5,6,7,8,9,10}"
]
},
"execution_count": 135,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dom(i10)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Was kann man mit Funktionen machen? Da Funktionen *nur* besondere Relationen sind, und Relationen *nur* besondere Mengen sind, kann man alle Mengen und Relationsoperatoren anwenden:"
]
},
{
"cell_type": "code",
"execution_count": 136,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{A} = \\{1,2,3\\}$\n",
"* $\\mathit{i10} = \\{(1\\mapsto 2),(2\\mapsto 3),(3\\mapsto 4),(4\\mapsto 5),(5\\mapsto 6),(6\\mapsto 7),(7\\mapsto 8),(8\\mapsto 9),(9\\mapsto 10),(10\\mapsto 11)\\}$\n",
"* $\\mathit{z} = 5$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
"\tA = {1,2,3}\n",
"\ti10 = {(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}\n",
"\tz = 5"
]
},
"execution_count": 136,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(4,z) ∈ i10"
]
},
{
"cell_type": "code",
"execution_count": 137,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,2,3,4,5,6,7,8,9,10\\}$"
],
"text/plain": [
"{1,2,3,4,5,6,7,8,9,10}"
]
},
"execution_count": 137,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dom(i10)"
]
},
{
"cell_type": "code",
"execution_count": 138,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{2,3,4,5,6,7,8,9,10,11\\}$"
],
"text/plain": [
"{2,3,4,5,6,7,8,9,10,11}"
]
},
"execution_count": 138,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ran(i10)"
]
},
{
"cell_type": "code",
"execution_count": 139,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{5,6\\}$"
],
"text/plain": [
"{5,6}"
]
},
"execution_count": 139,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"i10[{4,5}]"
]
},
{
"cell_type": "code",
"execution_count": 140,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{5\\}$"
],
"text/plain": [
"{5}"
]
},
"execution_count": 140,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"i10[{4}]"
]
},
{
"cell_type": "code",
"execution_count": 141,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{3\\}$"
],
"text/plain": [
"{3}"
]
},
"execution_count": 141,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"i10~[{4}]"
]
},
{
"cell_type": "code",
"execution_count": 142,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{6\\}$"
],
"text/plain": [
"{6}"
]
},
"execution_count": 142,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(i10 ; i10)[{4}]"
]
},
{
"cell_type": "code",
"execution_count": 143,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{5,6,7,8,9,10,11\\}$"
],
"text/plain": [
"{5,6,7,8,9,10,11}"
]
},
"execution_count": 143,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"closure1(i10)[{4}]"
]
},
{
"cell_type": "code",
"execution_count": 144,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,2,3\\}$"
],
"text/plain": [
"{1,2,3}"
]
},
"execution_count": 144,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"closure1(i10~)[{4}]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Anmerkung: um den Wert einer Funktion $F$ für eine Eingabe $x$ zu berechnen, kann man das relationale Bild $F[\\{x\\}]$ verwenden. Aber: das Ergebnis ist eine Menge! Man kann also nicht direkt zB arithmetische Operatoren anwenden. Dafür wird eine neue Operation für Funktionen eingeführt: die **Funktionsanwendung**: ```F(x)```."
]
},
{
"cell_type": "code",
"execution_count": 145,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$5$"
],
"text/plain": [
"5"
]
},
"execution_count": 145,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"i10(4)"
]
},
{
"cell_type": "code",
"execution_count": 146,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$10$"
],
"text/plain": [
"10"
]
},
"execution_count": 146,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"i10(4) + i10(4)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Es ist ein Fehler eine Funktion ausserhalb des Definitionsbereiches anzuwenden. Beim relationalen Bild ist dies erlaubt und man bekommt dort die leere Menge als Ausgabe."
]
},
{
"cell_type": "code",
"execution_count": 147,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$"
],
"text/plain": [
"∅"
]
},
"execution_count": 147,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"i10[{100}]"
]
},
{
"cell_type": "code",
"execution_count": 148,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto 2),(2\\mapsto 3),(3\\mapsto 4),(4\\mapsto 5),(5\\mapsto 6),(6\\mapsto 7),(7\\mapsto 8),(8\\mapsto 9),(9\\mapsto 10),(10\\mapsto 11)\\}$"
],
"text/plain": [
"{(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
"execution_count": 148,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"i10"
]
},
{
"cell_type": "code",
"execution_count": 149,
"metadata": {},
"outputs": [
{
"ename": "CommandExecutionException",
"evalue": ":eval: NOT-WELL-DEFINED: \nfunction applied outside of domain (#7): Function argument: 100, function value: {}\n ### Line: 2, Column: 0 until 8\n\n",
"output_type": "error",
"traceback": [
"\u001b[1m\u001b[31m:eval: NOT-WELL-DEFINED: \u001b[0m",
"\u001b[1m\u001b[31mfunction applied outside of domain (#7): Function argument: 100, function value: {}\u001b[0m",
"\u001b[1m\u001b[31m ### Line: 2, Column: 0 until 8\u001b[0m"
]
}
],
"source": [
"i10(100)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Alle möglichen Funktionen über einen Definitionsbereich ergeben wieder eine Menge:"
]
},
{
"cell_type": "code",
"execution_count": 150,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{\\{(1\\mapsto 1),(2\\mapsto 1)\\},\\{(1\\mapsto 1),(2\\mapsto 2)\\},\\{(1\\mapsto 2),(2\\mapsto 1)\\},\\{(1\\mapsto 2),(2\\mapsto 2)\\}\\}$"
],
"text/plain": [
"{{(1↦1),(2↦1)},{(1↦1),(2↦2)},{(1↦2),(2↦1)},{(1↦2),(2↦2)}}"
]
},
"execution_count": 150,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1..2 --> 1..2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Es gibt auch noch andere Arten an Funktionen:\n",
"* partielle Funktionen, +->\n",
"* injektive (umkehrbare) Funktionen >->\n",
"* surjektive Funktionen die den Wertebereich komplett abdecken -->>\n",
"* injektive und surjektive Funktionen >->>, auch Bijektionen gennant\n",
"\n",
"(Die Pfeilsymbole sind sehr spezifisch für die B Sprache. Wir werden diese nicht im Skript verwenden.)"
]
},
{
"cell_type": "code",
"execution_count": 151,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\{\\emptyset,\\{(1\\mapsto 1)\\},\\{(1\\mapsto 1),(2\\mapsto 1)\\},\\{(1\\mapsto 1),(2\\mapsto 2)\\},\\{(1\\mapsto 2)\\},\\{(1\\mapsto 2),(2\\mapsto 1)\\},\\{(1\\mapsto 2),(2\\mapsto 2)\\},\\{(2\\mapsto 1)\\},\\{(2\\mapsto 2)\\}\\}$"
],
"text/plain": [
"{∅,{(1↦1)},{(1↦1),(2↦1)},{(1↦1),(2↦2)},{(1↦2)},{(1↦2),(2↦1)},{(1↦2),(2↦2)},{(2↦1)},{(2↦2)}}"
]
},
"execution_count": 151,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1..2 +-> 1..2"
]
},
{
"cell_type": "code",
"execution_count": 152,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{A} = \\{1,2,3\\}$\n",
"* $\\mathit{i10} = \\{(1\\mapsto 2),(2\\mapsto 3),(3\\mapsto 4),(4\\mapsto 5),(5\\mapsto 6),(6\\mapsto 7),(7\\mapsto 8),(8\\mapsto 9),(9\\mapsto 10),(10\\mapsto 11)\\}$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
"\tA = {1,2,3}\n",
"\ti10 = {(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
"execution_count": 152,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(1..2 --> 1..2) <: (1..2 +-> 1..2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Folgende fünf Funktionen sind partiell und nicht total:"
]
},
{
"cell_type": "code",
"execution_count": 153,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\{\\emptyset,\\{(1\\mapsto 1)\\},\\{(1\\mapsto 2)\\},\\{(2\\mapsto 1)\\},\\{(2\\mapsto 2)\\}\\}$"
],
"text/plain": [
"{∅,{(1↦1)},{(1↦2)},{(2↦1)},{(2↦2)}}"
]
},
"execution_count": 153,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(1..2 +-> 1..2) \\ (1..2 --> 1..2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Das sind die totalen umkehrbaren Funktionen:"
]
},
{
"cell_type": "code",
"execution_count": 154,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{\\{(1\\mapsto 1),(2\\mapsto 2)\\},\\{(1\\mapsto 2),(2\\mapsto 1)\\}\\}$"
],
"text/plain": [
"{{(1↦1),(2↦2)},{(1↦2),(2↦1)}}"
]
},
"execution_count": 154,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1..2 >-> 1..2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In diesem Falle sind diese identisch zu den surjektiven Funktionen und den Bijektionen:"
]
},
{
"cell_type": "code",
"execution_count": 155,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{\\{(1\\mapsto 1),(2\\mapsto 2)\\},\\{(1\\mapsto 2),(2\\mapsto 1)\\}\\}$"
],
"text/plain": [
"{{(1↦1),(2↦2)},{(1↦2),(2↦1)}}"
]
},
"execution_count": 155,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1..2 -->> 1..2"
]
},
{
"cell_type": "code",
"execution_count": 156,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{\\{(1\\mapsto 1),(2\\mapsto 2)\\},\\{(1\\mapsto 2),(2\\mapsto 1)\\}\\}$"
],
"text/plain": [
"{{(1↦1),(2↦2)},{(1↦2),(2↦1)}}"
]
},
"execution_count": 156,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1..2 >->> 1..2"
]
},
{
"cell_type": "code",
"execution_count": 157,
"metadata": {},
"outputs": [],
"source": [
":unlet i10"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Endliche Folgen\n",
" \n",
"Es gibt verschiedene Schreibweisen für endliche Folgen.\n",
"Zum Beispiel $[1,2]$ ist eine Folge bestehend aus der Zahl 1 gefolgt von der Zahl 2.\n",
"\n",
"(Anmerkung: Im Skript werden wir eine andere Schreibweise verwenden. Zum Beispiel, $ab$ ist die\n",
"Folge bestehen aus dem Symbol $a$ gefolgt von $b$.\n",
"Diese Schreibweise ist zwar kompakter, aber nicht immer eindeutig und nicht für eine maschinelle Verarbeitung geeeignet.)\n",
"\n",
"Was unterscheidet Folgen von Mengen?\n",
"Wie kann man Folgen in Mengentheorie und Logik darstellen?\n",
"\n",
" \n",
"## Endliche Folgen vs Mengen\n",
" \n",
"Die Reihenfolge der Elemente ist wichtig:\n",
" * $[1,2] \\neq [2,1] $, während\n",
" * $\\{1,2\\} = \\{2,1\\} $\n",
"\n",
"Elemente können mehrfach auftauchen:\n",
" * $[1,1] \\neq [1] $, während\n",
" * $\\{1,1\\} = \\{1\\} $\n",
"\n",
"Wie kann man Folgen in Mengentheorie und Logik darstellen?"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Endliche Folgen mathematisch gesehen\n",
" \n",
"Eine (endliche) Folge $G$ von $A$ Elementen der Länge $n$ ist \n",
"* eine (totale) Funktion von $1..n$ nach $A$.\n",
"\n",
"Da eine totale Funktion eine Menge an Paare ist, haben wir zum Beispiel:"
]
},
{
"cell_type": "code",
"execution_count": 158,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto 22),(2\\mapsto 22),(3\\mapsto 33)\\}$"
],
"text/plain": [
"{(1↦22),(2↦22),(3↦33)}"
]
},
"execution_count": 158,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":let G [22,22,33]"
]
},
{
"cell_type": "code",
"execution_count": 159,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{(1\\mapsto 22),(2\\mapsto 22),(3\\mapsto 33)\\}$"
],
"text/plain": [
"{(1↦22),(2↦22),(3↦33)}"
]
},
"execution_count": 159,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"G"
]
},
{
"cell_type": "code",
"execution_count": 160,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
"<!-- Generated by graphviz version 2.44.1 (0)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
"<svg width=\"206pt\" height=\"131pt\"\n",
" viewBox=\"0.00 0.00 206.00 131.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 127)\">\n",
"<title>state</title>\n",
"<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-127 202,-127 202,4 -4,4\"/>\n",
"<!-- 3 -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>3</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-123 0,-123 0,-87 54,-87 54,-123\"/>\n",
"<text text-anchor=\"middle\" x=\"27\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
"</g>\n",
"<!-- 33 -->\n",
"<g id=\"node2\" class=\"node\">\n",
"<title>33</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-36 0,-36 0,0 54,0 54,-36\"/>\n",
"<text text-anchor=\"middle\" x=\"27\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">33</text>\n",
"</g>\n",
"<!-- 3->33 -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>3->33</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-86.8C27,-75.16 27,-59.55 27,-46.24\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-46.18 27,-36.18 23.5,-46.18 30.5,-46.18\"/>\n",
"<text text-anchor=\"middle\" x=\"33\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">G</text>\n",
"</g>\n",
"<!-- 2 -->\n",
"<g id=\"node3\" class=\"node\">\n",
"<title>2</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"126,-123 72,-123 72,-87 126,-87 126,-123\"/>\n",
"<text text-anchor=\"middle\" x=\"99\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
"</g>\n",
"<!-- 22 -->\n",
"<g id=\"node4\" class=\"node\">\n",
"<title>22</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"162,-36 108,-36 108,0 162,0 162,-36\"/>\n",
"<text text-anchor=\"middle\" x=\"135\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">22</text>\n",
"</g>\n",
"<!-- 2->22 -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>2->22</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M106.29,-86.8C111.31,-74.93 118.09,-58.93 123.8,-45.45\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"127.05,-46.75 127.73,-36.18 120.6,-44.02 127.05,-46.75\"/>\n",
"<text text-anchor=\"middle\" x=\"125\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">G</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g id=\"node5\" class=\"node\">\n",
"<title>1</title>\n",
"<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"198,-123 144,-123 144,-87 198,-87 198,-123\"/>\n",
"<text text-anchor=\"middle\" x=\"171\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
"</g>\n",
"<!-- 1->22 -->\n",
"<g id=\"edge3\" class=\"edge\">\n",
"<title>1->22</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M163.71,-86.8C158.69,-74.93 151.91,-58.93 146.2,-45.45\"/>\n",
"<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"149.4,-44.02 142.27,-36.18 142.95,-46.75 149.4,-44.02\"/>\n",
"<text text-anchor=\"middle\" x=\"161\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">G</text>\n",
"</g>\n",
"</g>\n",
"</svg>"
],
"text/plain": [
"<Dot visualization: expr_as_graph [AGA={1,2,3} & G={(1,22),(2,22),(3,33)}(\"G\",G)]>"
]
},
"execution_count": 160,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
":dot expr_as_graph (\"G\",G)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Es gilt also\n",
"* $G = [22,22,33] = \\{(1\\mapsto 22),(2\\mapsto 22),(3\\mapsto 33)\\}$\n",
" \n",
"Wir schreiben $\\mathit{G} \\in \\mathit{seq}$(ℤ)\n",
" oder aber auch $G \\in ℤ^*$ (mehr dazu später im Skript)\n",
" \n",
"Das n-te Element einer Folge $G$ ist einfach $G(n)$:"
]
},
{
"cell_type": "code",
"execution_count": 161,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$22$"
],
"text/plain": [
"22"
]
},
"execution_count": 161,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"G(2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Die Länge einer Folge (```size``` in B) ist einfach die Kardinalität der unterliegenden Relation:"
]
},
{
"cell_type": "code",
"execution_count": 162,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$3$"
],
"text/plain": [
"3"
]
},
"execution_count": 162,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"size(G)"
]
},
{
"cell_type": "code",
"execution_count": 163,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$3$"
],
"text/plain": [
"3"
]
},
"execution_count": 163,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"card(G)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Wir können alle anderen Menge und Relationsoperatoren auf Folgen anwenden:"
]
},
{
"cell_type": "code",
"execution_count": 164,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,2,3\\}$"
],
"text/plain": [
"{1,2,3}"
]
},
"execution_count": 164,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dom(G)"
]
},
{
"cell_type": "code",
"execution_count": 165,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{22,33\\}$"
],
"text/plain": [
"{22,33}"
]
},
"execution_count": 165,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ran(G)"
]
},
{
"cell_type": "code",
"execution_count": 166,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$\\{1,2\\}$"
],
"text/plain": [
"{1,2}"
]
},
"execution_count": 166,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"G~[{22}]"
]
},
{
"cell_type": "code",
"execution_count": 167,
"metadata": {},
"outputs": [],
"source": [
":unlet G"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Zusammenfassung Mengentheorie\n",
" \n",
"* Mengen, Notationen (per Prädikat)\n",
"* Potenzmenge, Menge von Mengen, $\\phi \\neq \\{ \\phi\\}$\n",
"* kartesisches Produkt, Relationen als Menge von Paaren/Tupeln\n",
"* Definitionsbereich, Wertebereich, Abbild, Umkehrrelation\n",
"* Transitive Hülle\n",
"* Funktionen\n",
"* Endliche Folgen als Funktion von $ℤ$ nach Wertebereich"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Lernziele von Kapitel 0\n",
"* logische Formeln verstehen und schreiben können\n",
"* logische Beweise verstehen: Wahrheitstabelle, Widerspruch, Deduktiver Beweis, Äuivalenzbeweis\n",
"* Mengenausdrücke verstehen und nach Logik übbersetzen können\n",
"* Problemstellungen nach Logik und Mengentheorie übersetzen können\n",
"* Relationen, Funktionen und Folgen in Mengendarstellung bearbeiten können"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "ProB 2",
"language": "prob",
"name": "prob2"
},
"language_info": {
"codemirror_mode": "prob2_jupyter_repl",
"file_extension": ".prob",
"mimetype": "text/x-prob2-jupyter-repl",
"name": "prob"
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},
"nbformat": 4,
"nbformat_minor": 2
}