From 5b8e374f712423fabf39c3bdbc63e0104688887c Mon Sep 17 00:00:00 2001
From: Michael Leuschel <leuschel@uni-duesseldorf.de>
Date: Thu, 16 Apr 2020 18:32:25 +0200
Subject: [PATCH] add first material about set theory
---
info4/kapitel-0/Logik.ipynb | 2 +-
info4/kapitel-0/Mengentheorie.ipynb | 587 ++++++++++++++++++++++++++++
2 files changed, 588 insertions(+), 1 deletion(-)
diff --git a/info4/kapitel-0/Logik.ipynb b/info4/kapitel-0/Logik.ipynb
index 7e97521..2984425 100644
--- a/info4/kapitel-0/Logik.ipynb
+++ b/info4/kapitel-0/Logik.ipynb
@@ -639,7 +639,7 @@
"\n",
"Die Bedeutung dieser Junktoren kann man in folgender Wahrheitstabelle zusammenfassen:\n",
"\n",
- "\n",
+ "<img src=\"./img/wahrheitstabelle.png\" width=\"600\">\n",
"\n",
"Wie berechnet man damit den Wahrheitswert einer Formel wie $(p \\vee (\\neg p \\wedge q)$?\n",
"\n",
diff --git a/info4/kapitel-0/Mengentheorie.ipynb b/info4/kapitel-0/Mengentheorie.ipynb
index 96025b7..9676745 100644
--- a/info4/kapitel-0/Mengentheorie.ipynb
+++ b/info4/kapitel-0/Mengentheorie.ipynb
@@ -12,6 +12,593 @@
"\n"
]
},
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Mengen\n",
+ " \n",
+ "\n",
+ "Fundamentale Idee der Mengentheorie: \n",
+ "* _\"The ability to regard any collection of objects as a single entity (i.e., as a set).\"_ (Keith Devlin. Joy of Sets.)\n",
+ "\n",
+ "In der Regel gibt es eine Domäne an \"Objekten\" aus 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",
+ "* $\\forall(a,x).(a\\not\\in x \\Leftrightarrow \\neg(a \\in x))$\n",
+ "\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 wenn 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": 2,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{FALSE}$"
+ ],
+ "text/plain": [
+ "FALSE"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "∅ = {1}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{FALSE}$"
+ ],
+ "text/plain": [
+ "FALSE"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{1} = {1,2}"
+ ]
+ },
+ {
+ "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": [
+ "{1,2} = {1,2}"
+ ]
+ },
+ {
+ "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": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{TRUE}$"
+ ],
+ "text/plain": [
+ "TRUE"
+ ]
+ },
+ "execution_count": 5,
+ "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, die oft mit runden und eckigen Klammern geschriben werden:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{FALSE}$"
+ ],
+ "text/plain": [
+ "FALSE"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "(2,5,3) = (2,3,5)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{TRUE}$"
+ ],
+ "text/plain": [
+ "TRUE"
+ ]
+ },
+ "execution_count": 8,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "[2,5,3] = [2,5,3]"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "* Elemente können in der Enumeration mehrfach auftauchen:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{TRUE}$"
+ ],
+ "text/plain": [
+ "TRUE"
+ ]
+ },
+ "execution_count": 6,
+ "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": 10,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{2,3,5,7\\}$"
+ ],
+ "text/plain": [
+ "{2,3,5,7}"
+ ]
+ },
+ "execution_count": 10,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{2,3,5} ∪ {5,7}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{5\\}$"
+ ],
+ "text/plain": [
+ "{5}"
+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{2,3,5}∩{5,7}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{2,3\\}$"
+ ],
+ "text/plain": [
+ "{2,3}"
+ ]
+ },
+ "execution_count": 12,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{2,3,5}−{5,7}"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Notationen für Mengen: per Prädikat\n",
+ " \n",
+ "Definition der Elemente durch ein Prädikat $\\{a \\mid P(a)\\}$:\n",
+ "* $z = \\{a \\mid P(a)\\} \\Leftrightarrow \\forall(a).(a\\in z \\equiv P(a))$\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{2,3,5\\}$"
+ ],
+ "text/plain": [
+ "{2,3,5}"
+ ]
+ },
+ "execution_count": 13,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{a | a>1 & a<6 & a≠4}"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Man kann damit auch unendliche Mengen darstellen:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{\\mathit{a}\\mid \\mathit{a} > 10\\}$"
+ ],
+ "text/plain": [
+ "{a∣a > 10}"
+ ]
+ },
+ "execution_count": 14,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{a | a>10}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 15,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{2\\}$"
+ ],
+ "text/plain": [
+ "{2}"
+ ]
+ },
+ "execution_count": 15,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{a | a mod 2 = 0} ∩ {2,3,5}"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Man kann damit 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": 16,
+ "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": 16,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "x = {2,3} & y = {2,5} & xy = {a| a∈x ∨ a∈y}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 17,
+ "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": 17,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "x = {2,3} & y = {2,5} & xy = {a| a∈x ∧ a∈y}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 18,
+ "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": 18,
+ "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": 19,
+ "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": 19,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "1..10"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Ein Theorem\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.pdf\" width=\"300\">"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 24,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{2,3,44,55\\}$"
+ ],
+ "text/plain": [
+ "{2,3,44,55}"
+ ]
+ },
+ "execution_count": 24,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{2,3,55}∪({2,44,77}∩{2,44,66})"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 25,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{2,3,44,55\\}$"
+ ],
+ "text/plain": [
+ "{2,3,44,55}"
+ ]
+ },
+ "execution_count": 25,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "({2,3,55}∪{2,44,77})∩({2,3,55}∪{2,44,66})"
+ ]
+ },
{
"cell_type": "code",
"execution_count": null,
--
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