diff --git a/info4/kapitel-0/Mengentheorie.ipynb b/info4/kapitel-0/Mengentheorie.ipynb
index f09b27b59256c3fa628618b2a969093410373988..130e993a01f26bb83f23d0956d18c416ec358092 100644
--- a/info4/kapitel-0/Mengentheorie.ipynb
+++ b/info4/kapitel-0/Mengentheorie.ipynb
@@ -29,9 +29,15 @@
"# Mengen\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. Springer-Verlag.)\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",
@@ -62,7 +68,7 @@
},
{
"cell_type": "code",
- "execution_count": 2,
+ "execution_count": 17,
"metadata": {},
"outputs": [
{
@@ -74,7 +80,7 @@
"FALSE"
]
},
- "execution_count": 2,
+ "execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
@@ -85,7 +91,7 @@
},
{
"cell_type": "code",
- "execution_count": 3,
+ "execution_count": 18,
"metadata": {},
"outputs": [
{
@@ -97,7 +103,7 @@
"FALSE"
]
},
- "execution_count": 3,
+ "execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
@@ -108,7 +114,7 @@
},
{
"cell_type": "code",
- "execution_count": 1,
+ "execution_count": 19,
"metadata": {},
"outputs": [
{
@@ -120,7 +126,7 @@
"TRUE"
]
},
- "execution_count": 1,
+ "execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
@@ -141,7 +147,7 @@
},
{
"cell_type": "code",
- "execution_count": 5,
+ "execution_count": 20,
"metadata": {},
"outputs": [
{
@@ -153,7 +159,7 @@
"TRUE"
]
},
- "execution_count": 5,
+ "execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
@@ -171,7 +177,7 @@
},
{
"cell_type": "code",
- "execution_count": 7,
+ "execution_count": 22,
"metadata": {},
"outputs": [
{
@@ -183,18 +189,18 @@
"FALSE"
]
},
- "execution_count": 7,
+ "execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
- "(2,5,3) = (2,3,5) // Vergleich von zwei Tripeln"
+ "(TRUE,5,3) = (TRUE,3,5) // Vergleich von zwei Tripeln"
]
},
{
"cell_type": "code",
- "execution_count": 2,
+ "execution_count": 23,
"metadata": {},
"outputs": [
{
@@ -206,7 +212,7 @@
"FALSE"
]
},
- "execution_count": 2,
+ "execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
@@ -224,7 +230,7 @@
},
{
"cell_type": "code",
- "execution_count": 6,
+ "execution_count": 24,
"metadata": {},
"outputs": [
{
@@ -236,7 +242,7 @@
"TRUE"
]
},
- "execution_count": 6,
+ "execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
@@ -265,7 +271,7 @@
},
{
"cell_type": "code",
- "execution_count": 10,
+ "execution_count": 25,
"metadata": {},
"outputs": [
{
@@ -277,7 +283,7 @@
"{2,3,5,7}"
]
},
- "execution_count": 10,
+ "execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
@@ -288,7 +294,7 @@
},
{
"cell_type": "code",
- "execution_count": 11,
+ "execution_count": 26,
"metadata": {},
"outputs": [
{
@@ -300,7 +306,7 @@
"{5}"
]
},
- "execution_count": 11,
+ "execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
@@ -311,7 +317,7 @@
},
{
"cell_type": "code",
- "execution_count": 12,
+ "execution_count": 27,
"metadata": {},
"outputs": [
{
@@ -323,7 +329,7 @@
"{2,3}"
]
},
- "execution_count": 12,
+ "execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
@@ -340,12 +346,12 @@
" \n",
"Man kann die Elemente einer Menge auch durch ein Prädikat beschreiben:$\\{a \\mid P(a)\\}$.\n",
"Es gilt:\n",
- "* $z = \\{a \\mid P(a)\\} \\Leftrightarrow \\forall(a).(a\\in z \\equiv P(a))$\n"
+ "* $z = \\{a \\mid P(a)\\} \\Leftrightarrow \\forall(a).(a\\in z \\Leftrightarrow P(a))$\n"
]
},
{
"cell_type": "code",
- "execution_count": 13,
+ "execution_count": 28,
"metadata": {},
"outputs": [
{
@@ -357,7 +363,7 @@
"{2,3,5}"
]
},
- "execution_count": 13,
+ "execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
@@ -375,7 +381,7 @@
},
{
"cell_type": "code",
- "execution_count": 14,
+ "execution_count": 29,
"metadata": {},
"outputs": [
{
@@ -387,7 +393,7 @@
"{a∣a > 10}"
]
},
- "execution_count": 14,
+ "execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
@@ -398,7 +404,7 @@
},
{
"cell_type": "code",
- "execution_count": 15,
+ "execution_count": 30,
"metadata": {},
"outputs": [
{
@@ -410,7 +416,7 @@
"{2}"
]
},
- "execution_count": 15,
+ "execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
@@ -432,7 +438,7 @@
},
{
"cell_type": "code",
- "execution_count": 16,
+ "execution_count": 31,
"metadata": {},
"outputs": [
{
@@ -454,7 +460,7 @@
"\ty = {2,5}"
]
},
- "execution_count": 16,
+ "execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
@@ -465,7 +471,7 @@
},
{
"cell_type": "code",
- "execution_count": 17,
+ "execution_count": 32,
"metadata": {},
"outputs": [
{
@@ -487,7 +493,7 @@
"\ty = {2,5}"
]
},
- "execution_count": 17,
+ "execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
@@ -498,7 +504,7 @@
},
{
"cell_type": "code",
- "execution_count": 18,
+ "execution_count": 33,
"metadata": {},
"outputs": [
{
@@ -520,7 +526,7 @@
"\ty = {2,5}"
]
},
- "execution_count": 18,
+ "execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
@@ -538,7 +544,7 @@
},
{
"cell_type": "code",
- "execution_count": 19,
+ "execution_count": 34,
"metadata": {},
"outputs": [
{
@@ -550,7 +556,7 @@
"{1,2,3,4,5,6,7,8,9,10}"
]
},
- "execution_count": 19,
+ "execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
@@ -573,7 +579,7 @@
},
{
"cell_type": "code",
- "execution_count": 24,
+ "execution_count": 35,
"metadata": {},
"outputs": [
{
@@ -585,7 +591,7 @@
"{2,3,44,55}"
]
},
- "execution_count": 24,
+ "execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
@@ -596,7 +602,7 @@
},
{
"cell_type": "code",
- "execution_count": 25,
+ "execution_count": 36,
"metadata": {},
"outputs": [
{
@@ -608,7 +614,7 @@
"{2,3,44,55}"
]
},
- "execution_count": 25,
+ "execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
@@ -630,14 +636,14 @@
"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. $x \\cup (y \\cap z)$\n",
- "2. (Lemma 1) $\\Longleftrightarrow$ $\\{a \\mid a\\in x \\vee a\\in (y \\cap z)\\}$ \n",
- "3. (Lemma 2) $\\Longleftrightarrow$ $\\{a \\mid a\\in x \\vee a\\in \\{b \\mid b\\in y \\wedge b\\in z\\} \\}$ \n",
- "4. (Lemma 3) $\\Longleftrightarrow$ $\\{a \\mid a\\in x \\vee (a\\in y \\wedge a\\in z) \\}$\n",
- "5. (Lemma 4) $\\Longleftrightarrow$ $\\{a \\mid (a\\in x \\vee a\\in y) \\wedge (a\\in x \\vee a\\in z) \\}$\n",
- "6. (Lemma 3, $\\Leftarrow$) $\\Longleftrightarrow$ $\\{a \\mid a\\in \\{b \\mid b\\in x \\vee b\\in y\\} \\wedge a\\in \\{b \\mid b\\in x \\vee b\\in z\\} \\}$\n",
- "7. (Lemma 1, $\\Leftarrow$) $\\Longleftrightarrow$ $\\{a \\mid a\\in x \\cup y \\wedge a\\in x \\cup z) \\}$\n",
- "8. (Lemma 2, $\\Leftarrow$) $(x\\cup y) \\cap (x\\cup z)$"
+ "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, $\\Leftarrow$) $m=(x\\cup y) \\cap (x\\cup z)$"
]
},
{
@@ -662,7 +668,7 @@
},
{
"cell_type": "code",
- "execution_count": 31,
+ "execution_count": 37,
"metadata": {},
"outputs": [
{
@@ -674,7 +680,7 @@
"{1,3,7,9}"
]
},
- "execution_count": 31,
+ "execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
@@ -685,7 +691,7 @@
},
{
"cell_type": "code",
- "execution_count": 28,
+ "execution_count": 38,
"metadata": {},
"outputs": [
{
@@ -697,7 +703,7 @@
"{1,3,5,7,9,10}"
]
},
- "execution_count": 28,
+ "execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
@@ -708,7 +714,7 @@
},
{
"cell_type": "code",
- "execution_count": 29,
+ "execution_count": 39,
"metadata": {},
"outputs": [
{
@@ -720,7 +726,7 @@
"{1,2,3,4,6,7,8,9}"
]
},
- "execution_count": 29,
+ "execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
@@ -731,7 +737,7 @@
},
{
"cell_type": "code",
- "execution_count": 32,
+ "execution_count": 40,
"metadata": {},
"outputs": [
{
@@ -743,7 +749,7 @@
"{1,3,7,9}"
]
},
- "execution_count": 32,
+ "execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
@@ -773,23 +779,9 @@
},
{
"cell_type": "code",
- "execution_count": 37,
+ "execution_count": null,
"metadata": {},
- "outputs": [
- {
- "data": {
- "text/markdown": [
- "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$"
- ],
- "text/plain": [
- "∅"
- ]
- },
- "execution_count": 37,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
+ "outputs": [],
"source": [
"{2,4} ∩ (ℤ \\ {2,4})"
]
@@ -813,7 +805,7 @@
},
{
"cell_type": "code",
- "execution_count": 68,
+ "execution_count": 41,
"metadata": {},
"outputs": [
{
@@ -833,7 +825,7 @@
"\ty = (1 ‥ 5)"
]
},
- "execution_count": 68,
+ "execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
@@ -844,7 +836,7 @@
},
{
"cell_type": "code",
- "execution_count": 67,
+ "execution_count": 42,
"metadata": {},
"outputs": [
{
@@ -856,7 +848,7 @@
"FALSE"
]
},
- "execution_count": 67,
+ "execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
@@ -867,7 +859,7 @@
},
{
"cell_type": "code",
- "execution_count": 61,
+ "execution_count": 43,
"metadata": {},
"outputs": [
{
@@ -879,7 +871,7 @@
"TRUE"
]
},
- "execution_count": 61,
+ "execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
@@ -900,7 +892,7 @@
},
{
"cell_type": "code",
- "execution_count": 38,
+ "execution_count": 44,
"metadata": {},
"outputs": [
{
@@ -912,7 +904,7 @@
"3"
]
},
- "execution_count": 38,
+ "execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
@@ -923,7 +915,7 @@
},
{
"cell_type": "code",
- "execution_count": 39,
+ "execution_count": 45,
"metadata": {},
"outputs": [
{
@@ -935,7 +927,7 @@
"3"
]
},
- "execution_count": 39,
+ "execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
@@ -946,7 +938,7 @@
},
{
"cell_type": "code",
- "execution_count": 40,
+ "execution_count": 46,
"metadata": {},
"outputs": [
{
@@ -958,7 +950,7 @@
"0"
]
},
- "execution_count": 40,
+ "execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
@@ -976,7 +968,7 @@
},
{
"cell_type": "code",
- "execution_count": 41,
+ "execution_count": 47,
"metadata": {},
"outputs": [
{
@@ -1023,7 +1015,7 @@
},
{
"cell_type": "code",
- "execution_count": 69,
+ "execution_count": 48,
"metadata": {},
"outputs": [
{
@@ -1055,7 +1047,7 @@
"\tO = 0"
]
},
- "execution_count": 69,
+ "execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
@@ -1073,7 +1065,7 @@
},
{
"cell_type": "code",
- "execution_count": 43,
+ "execution_count": 49,
"metadata": {},
"outputs": [
{
@@ -1082,30 +1074,30 @@
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
- "* $\\mathit{R} = 1$\n",
- "* $\\mathit{S} = 7$\n",
- "* $\\mathit{D} = 4$\n",
- "* $\\mathit{E} = 6$\n",
- "* $\\mathit{Y} = 0$\n",
- "* $\\mathit{M} = 3$\n",
- "* $\\mathit{N} = 5$\n",
- "* $\\mathit{O} = 2$"
+ "* $\\mathit{R} = 2$\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 = 1\n",
- "\tS = 7\n",
- "\tD = 4\n",
- "\tE = 6\n",
- "\tY = 0\n",
- "\tM = 3\n",
- "\tN = 5\n",
- "\tO = 2"
+ "\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": 43,
+ "execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
@@ -1124,7 +1116,7 @@
},
{
"cell_type": "code",
- "execution_count": 45,
+ "execution_count": 50,
"metadata": {},
"outputs": [
{
@@ -1133,37 +1125,37 @@
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
- "* $\\mathit{R} = 1$\n",
- "* $\\mathit{S} = 7$\n",
- "* $\\mathit{D} = 4$\n",
- "* $\\mathit{E} = 6$\n",
+ "* $\\mathit{R} = 3$\n",
+ "* $\\mathit{S} = 2$\n",
+ "* $\\mathit{D} = 5$\n",
+ "* $\\mathit{E} = 7$\n",
"* $\\mathit{Y} = 0$\n",
- "* $\\mathit{M} = 3$\n",
- "* $\\mathit{N} = 5$\n",
- "* $\\mathit{O} = 2$"
+ "* $\\mathit{M} = 1$\n",
+ "* $\\mathit{N} = 6$\n",
+ "* $\\mathit{O} = 4$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
- "\tR = 1\n",
- "\tS = 7\n",
- "\tD = 4\n",
- "\tE = 6\n",
+ "\tR = 3\n",
+ "\tS = 2\n",
+ "\tD = 5\n",
+ "\tE = 7\n",
"\tY = 0\n",
- "\tM = 3\n",
- "\tN = 5\n",
- "\tO = 2"
+ "\tM = 1\n",
+ "\tN = 6\n",
+ "\tO = 4"
]
},
- "execution_count": 45,
+ "execution_count": 50,
"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"
+ "S ≠ 0 ∧ M ≠ 0"
]
},
{
@@ -1175,7 +1167,7 @@
},
{
"cell_type": "code",
- "execution_count": 70,
+ "execution_count": 51,
"metadata": {},
"outputs": [
{
@@ -1207,7 +1199,7 @@
"\tO = 0"
]
},
- "execution_count": 70,
+ "execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
@@ -1228,7 +1220,7 @@
},
{
"cell_type": "code",
- "execution_count": 48,
+ "execution_count": 52,
"metadata": {},
"outputs": [
{
@@ -1240,7 +1232,7 @@
"{(9↦5↦6↦7↦1↦0↦8↦2)}"
]
},
- "execution_count": 48,
+ "execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
@@ -1256,7 +1248,7 @@
},
{
"cell_type": "code",
- "execution_count": 71,
+ "execution_count": 53,
"metadata": {},
"outputs": [
{
@@ -1271,7 +1263,7 @@
"9\t5\t6\t7\t1\t0\t8\t2\n"
]
},
- "execution_count": 71,
+ "execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
@@ -1308,7 +1300,7 @@
},
{
"cell_type": "code",
- "execution_count": 72,
+ "execution_count": 54,
"metadata": {},
"outputs": [
{
@@ -1323,7 +1315,7 @@
"2\t0\t3\t4\t1\t8\t9\n"
]
},
- "execution_count": 72,
+ "execution_count": 54,
"metadata": {},
"output_type": "execute_result"
}
@@ -1361,7 +1353,7 @@
},
{
"cell_type": "code",
- "execution_count": 57,
+ "execution_count": 55,
"metadata": {},
"outputs": [
{
@@ -1373,7 +1365,7 @@
"3"
]
},
- "execution_count": 57,
+ "execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
@@ -1384,7 +1376,7 @@
},
{
"cell_type": "code",
- "execution_count": 52,
+ "execution_count": 56,
"metadata": {},
"outputs": [
{
@@ -1396,7 +1388,7 @@
"1"
]
},
- "execution_count": 52,
+ "execution_count": 56,
"metadata": {},
"output_type": "execute_result"
}
@@ -1407,7 +1399,7 @@
},
{
"cell_type": "code",
- "execution_count": 53,
+ "execution_count": 57,
"metadata": {},
"outputs": [
{
@@ -1419,7 +1411,7 @@
"2"
]
},
- "execution_count": 53,
+ "execution_count": 57,
"metadata": {},
"output_type": "execute_result"
}
@@ -1430,7 +1422,7 @@
},
{
"cell_type": "code",
- "execution_count": 55,
+ "execution_count": 58,
"metadata": {},
"outputs": [
{
@@ -1442,7 +1434,7 @@
"TRUE"
]
},
- "execution_count": 55,
+ "execution_count": 58,
"metadata": {},
"output_type": "execute_result"
}
@@ -1453,7 +1445,7 @@
},
{
"cell_type": "code",
- "execution_count": 56,
+ "execution_count": 59,
"metadata": {},
"outputs": [
{
@@ -1465,7 +1457,7 @@
"FALSE"
]
},
- "execution_count": 56,
+ "execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
@@ -1486,7 +1478,7 @@
},
{
"cell_type": "code",
- "execution_count": 76,
+ "execution_count": 60,
"metadata": {},
"outputs": [
{
@@ -1498,7 +1490,7 @@
"FALSE"
]
},
- "execution_count": 76,
+ "execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
@@ -1509,7 +1501,7 @@
},
{
"cell_type": "code",
- "execution_count": 77,
+ "execution_count": 61,
"metadata": {},
"outputs": [
{
@@ -1521,7 +1513,7 @@
"0"
]
},
- "execution_count": 77,
+ "execution_count": 61,
"metadata": {},
"output_type": "execute_result"
}
@@ -1532,7 +1524,7 @@
},
{
"cell_type": "code",
- "execution_count": 78,
+ "execution_count": 62,
"metadata": {},
"outputs": [
{
@@ -1544,7 +1536,7 @@
"1"
]
},
- "execution_count": 78,
+ "execution_count": 62,
"metadata": {},
"output_type": "execute_result"
}
@@ -1555,7 +1547,7 @@
},
{
"cell_type": "code",
- "execution_count": 83,
+ "execution_count": 63,
"metadata": {},
"outputs": [
{
@@ -1567,7 +1559,7 @@
"TRUE"
]
},
- "execution_count": 83,
+ "execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
@@ -1578,7 +1570,7 @@
},
{
"cell_type": "code",
- "execution_count": 84,
+ "execution_count": 64,
"metadata": {},
"outputs": [
{
@@ -1590,7 +1582,7 @@
"FALSE"
]
},
- "execution_count": 84,
+ "execution_count": 64,
"metadata": {},
"output_type": "execute_result"
}
@@ -1601,7 +1593,7 @@
},
{
"cell_type": "code",
- "execution_count": 85,
+ "execution_count": 65,
"metadata": {},
"outputs": [
{
@@ -1613,7 +1605,7 @@
"TRUE"
]
},
- "execution_count": 85,
+ "execution_count": 65,
"metadata": {},
"output_type": "execute_result"
}
@@ -1632,7 +1624,7 @@
},
{
"cell_type": "code",
- "execution_count": 7,
+ "execution_count": 66,
"metadata": {},
"outputs": [
{
@@ -1644,7 +1636,7 @@
"{∅,{1},{1,2},{2}}"
]
},
- "execution_count": 7,
+ "execution_count": 66,
"metadata": {},
"output_type": "execute_result"
}
@@ -1655,7 +1647,7 @@
},
{
"cell_type": "code",
- "execution_count": 8,
+ "execution_count": 67,
"metadata": {},
"outputs": [
{
@@ -1667,7 +1659,7 @@
"{∅}"
]
},
- "execution_count": 8,
+ "execution_count": 67,
"metadata": {},
"output_type": "execute_result"
}
@@ -1678,36 +1670,30 @@
},
{
"cell_type": "code",
- "execution_count": 9,
+ "execution_count": 68,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
- "$\\newcommand{\\pow}{\\mathop{\\mathbb P\\hbox{}}\\nolimits}\\newcommand{\\upto}{\\mathbin{.\\mkern1mu.}}\\mathit{TRUE}$\n",
- "\n",
- "**Solution:**\n",
- "* $\\mathit{x} = \\pow((1 \\upto 3))$"
+ "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\{\\emptyset,\\{1\\},\\{1,2\\},\\{1,3\\},\\{2\\},\\{1,2,3\\},\\{2,3\\},\\{3\\}\\}$"
],
"text/plain": [
- "TRUE\n",
- "\n",
- "Solution:\n",
- "\tx = ℙ((1 ‥ 3))"
+ "{∅,{1},{1,2},{1,3},{2},{1,2,3},{2,3},{3}}"
]
},
- "execution_count": 9,
+ "execution_count": 68,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
- "ℙ(1..3)=x"
+ "ℙ(1..3)"
]
},
{
"cell_type": "code",
- "execution_count": 10,
+ "execution_count": 69,
"metadata": {},
"outputs": [
{
@@ -1736,7 +1722,7 @@
"{3}\n"
]
},
- "execution_count": 10,
+ "execution_count": 69,
"metadata": {},
"output_type": "execute_result"
}
@@ -1747,7 +1733,7 @@
},
{
"cell_type": "code",
- "execution_count": 11,
+ "execution_count": 70,
"metadata": {},
"outputs": [
{
@@ -1768,7 +1754,7 @@
"{{1}}\n"
]
},
- "execution_count": 11,
+ "execution_count": 70,
"metadata": {},
"output_type": "execute_result"
}
@@ -1779,7 +1765,7 @@
},
{
"cell_type": "code",
- "execution_count": 12,
+ "execution_count": 71,
"metadata": {},
"outputs": [
{
@@ -1791,7 +1777,7 @@
"8"
]
},
- "execution_count": 12,
+ "execution_count": 71,
"metadata": {},
"output_type": "execute_result"
}
@@ -1802,7 +1788,7 @@
},
{
"cell_type": "code",
- "execution_count": 13,
+ "execution_count": 72,
"metadata": {},
"outputs": [
{
@@ -1814,7 +1800,7 @@
"1024"
]
},
- "execution_count": 13,
+ "execution_count": 72,
"metadata": {},
"output_type": "execute_result"
}
@@ -1825,7 +1811,7 @@
},
{
"cell_type": "code",
- "execution_count": 14,
+ "execution_count": 73,
"metadata": {},
"outputs": [
{
@@ -1837,7 +1823,7 @@
"1267650600228229401496703205376"
]
},
- "execution_count": 14,
+ "execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
@@ -1848,7 +1834,7 @@
},
{
"cell_type": "code",
- "execution_count": 15,
+ "execution_count": 74,
"metadata": {},
"outputs": [
{
@@ -1860,7 +1846,7 @@
"340282366920938463463374607431768211456"
]
},
- "execution_count": 15,
+ "execution_count": 74,
"metadata": {},
"output_type": "execute_result"
}
@@ -1871,7 +1857,7 @@
},
{
"cell_type": "code",
- "execution_count": 16,
+ "execution_count": 75,
"metadata": {},
"outputs": [
{
@@ -1883,7 +1869,7 @@
"115792089237316195423570985008687907853269984665640564039457584007913129639936"
]
},
- "execution_count": 16,
+ "execution_count": 75,
"metadata": {},
"output_type": "execute_result"
}
@@ -1911,7 +1897,6 @@
"\n",
"Eine unäre Relation über $x$ entspricht einfach einer Untermenge von $x$:\n",
"* die Menge an Werten für die die Relation wahr ist.\n",
- " \\end{itemize}\n",
"\n",
"Beispiele:\n",
"* Relation ``Ziffer'' über die ganzen Zahlen ist\n",
@@ -1919,10 +1904,16 @@
"* 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",
+ " * $\\{ \\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",
- "Relationen ist die explizite Darstellung eines Prädikats als Menge:\n",
+ "Eine Relation ist die explizite Darstellung eines Prädikats als Menge:\n",
" \n",
"* Prädikate können mit logischen Junktoren und Quantoren verarabeitet werden:\n",
" * $\\exists x. is\\_rich(x)$, $\\forall x.(is\\_poor(x) \\Rightarrow \\neg is\\_rich(x))$\n",
@@ -1947,7 +1938,7 @@
},
{
"cell_type": "code",
- "execution_count": 102,
+ "execution_count": 78,
"metadata": {},
"outputs": [
{
@@ -1959,7 +1950,7 @@
"{(1↦4),(1↦5),(2↦4),(2↦5)}"
]
},
- "execution_count": 102,
+ "execution_count": 78,
"metadata": {},
"output_type": "execute_result"
}
@@ -1970,7 +1961,7 @@
},
{
"cell_type": "code",
- "execution_count": 103,
+ "execution_count": 79,
"metadata": {},
"outputs": [
{
@@ -1982,7 +1973,7 @@
"{(1↦4),(2↦4)}"
]
},
- "execution_count": 103,
+ "execution_count": 79,
"metadata": {},
"output_type": "execute_result"
}
@@ -1993,7 +1984,7 @@
},
{
"cell_type": "code",
- "execution_count": 104,
+ "execution_count": 80,
"metadata": {},
"outputs": [
{
@@ -2005,7 +1996,7 @@
"{(1↦FALSE),(1↦TRUE),(2↦FALSE),(2↦TRUE)}"
]
},
- "execution_count": 104,
+ "execution_count": 80,
"metadata": {},
"output_type": "execute_result"
}
@@ -2016,7 +2007,7 @@
},
{
"cell_type": "code",
- "execution_count": 105,
+ "execution_count": 81,
"metadata": {},
"outputs": [
{
@@ -2028,7 +2019,7 @@
"{(1↦1),(1↦2),(1↦3),(2↦1),(2↦2),(2↦3),(3↦1),(3↦2),(3↦3)}"
]
},
- "execution_count": 105,
+ "execution_count": 81,
"metadata": {},
"output_type": "execute_result"
}
@@ -2039,7 +2030,7 @@
},
{
"cell_type": "code",
- "execution_count": 106,
+ "execution_count": 82,
"metadata": {},
"outputs": [
{
@@ -2051,7 +2042,7 @@
"100"
]
},
- "execution_count": 106,
+ "execution_count": 82,
"metadata": {},
"output_type": "execute_result"
}
@@ -2069,7 +2060,7 @@
},
{
"cell_type": "code",
- "execution_count": 107,
+ "execution_count": 83,
"metadata": {},
"outputs": [
{
@@ -2081,7 +2072,7 @@
"{(2↦3)}"
]
},
- "execution_count": 107,
+ "execution_count": 83,
"metadata": {},
"output_type": "execute_result"
}
@@ -2092,7 +2083,7 @@
},
{
"cell_type": "code",
- "execution_count": 108,
+ "execution_count": 86,
"metadata": {},
"outputs": [
{
@@ -2104,7 +2095,7 @@
"{(3↦2)}"
]
},
- "execution_count": 108,
+ "execution_count": 86,
"metadata": {},
"output_type": "execute_result"
}
@@ -2135,25 +2126,25 @@
},
{
"cell_type": "code",
- "execution_count": 113,
+ "execution_count": 87,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
- "$\\{(1\\mapsto 2),(1\\mapsto 3),(1\\mapsto 4),(1\\mapsto 5),(1\\mapsto 6),(1\\mapsto 7),(1\\mapsto 8),(1\\mapsto 9),(2\\mapsto 3),(2\\mapsto 4),(2\\mapsto 5),(2\\mapsto 6),(2\\mapsto 7),(2\\mapsto 8),(2\\mapsto 9),(3\\mapsto 4),(3\\mapsto 5),(3\\mapsto 6),(3\\mapsto 7),(3\\mapsto 8),(3\\mapsto 9),(4\\mapsto 5),(4\\mapsto 6),(4\\mapsto 7),(4\\mapsto 8),(4\\mapsto 9),(5\\mapsto 6),(5\\mapsto 7),(5\\mapsto 8),(5\\mapsto 9),(6\\mapsto 7),(6\\mapsto 8),(6\\mapsto 9),(7\\mapsto 8),(7\\mapsto 9),(8\\mapsto 9)\\}$"
+ "$\\{(0\\mapsto 1),(0\\mapsto 2),(0\\mapsto 3),(0\\mapsto 4),(0\\mapsto 5),(0\\mapsto 6),(0\\mapsto 7),(0\\mapsto 8),(0\\mapsto 9),(1\\mapsto 2),(1\\mapsto 3),(1\\mapsto 4),(1\\mapsto 5),(1\\mapsto 6),(1\\mapsto 7),(1\\mapsto 8),(1\\mapsto 9),(2\\mapsto 3),(2\\mapsto 4),(2\\mapsto 5),(2\\mapsto 6),(2\\mapsto 7),(2\\mapsto 8),(2\\mapsto 9),(3\\mapsto 4),(3\\mapsto 5),(3\\mapsto 6),(3\\mapsto 7),(3\\mapsto 8),(3\\mapsto 9),(4\\mapsto 5),(4\\mapsto 6),(4\\mapsto 7),(4\\mapsto 8),(4\\mapsto 9),(5\\mapsto 6),(5\\mapsto 7),(5\\mapsto 8),(5\\mapsto 9),(6\\mapsto 7),(6\\mapsto 8),(6\\mapsto 9),(7\\mapsto 8),(7\\mapsto 9),(8\\mapsto 9)\\}$"
],
"text/plain": [
- "{(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)}"
+ "{(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": 113,
+ "execution_count": 87,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
- "{a,b| (a,b)∈(1..9)×(1..9) ∧ a<b}"
+ "{a,b| (a,b)∈(0..9)×(0..9) ∧ a<b}"
]
},
{
@@ -2166,7 +2157,7 @@
},
{
"cell_type": "code",
- "execution_count": 116,
+ "execution_count": 88,
"metadata": {},
"outputs": [
{
@@ -2178,7 +2169,7 @@
"{(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)}"
]
},
- "execution_count": 116,
+ "execution_count": 88,
"metadata": {},
"output_type": "execute_result"
}
@@ -2196,7 +2187,7 @@
},
{
"cell_type": "code",
- "execution_count": 119,
+ "execution_count": 89,
"metadata": {},
"outputs": [
{
@@ -2290,7 +2281,7 @@
"<Dot visualization: expr_as_graph [(\"halb\",{a,b|a:1..10 & b:1..10 & b*2=a})]>"
]
},
- "execution_count": 119,
+ "execution_count": 89,
"metadata": {},
"output_type": "execute_result"
}
@@ -2309,7 +2300,7 @@
},
{
"cell_type": "code",
- "execution_count": 123,
+ "execution_count": 90,
"metadata": {},
"outputs": [
{
@@ -2321,7 +2312,7 @@
"{(1↦\"nein\"),(2↦\"nein\"),(3↦\"ja\"),(4↦\"nein\"),(5↦\"nein\"),(6↦\"ja\"),(7↦\"nein\")}"
]
},
- "execution_count": 123,
+ "execution_count": 90,
"metadata": {},
"output_type": "execute_result"
}
@@ -2332,7 +2323,7 @@
},
{
"cell_type": "code",
- "execution_count": 124,
+ "execution_count": 91,
"metadata": {},
"outputs": [
{
@@ -2443,7 +2434,7 @@
"<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": 124,
+ "execution_count": 91,
"metadata": {},
"output_type": "execute_result"
}
@@ -2461,7 +2452,7 @@
},
{
"cell_type": "code",
- "execution_count": 142,
+ "execution_count": 92,
"metadata": {},
"outputs": [
{
@@ -2470,7 +2461,7 @@
"Preference changed: DOT_ENGINE = circo\n"
]
},
- "execution_count": 142,
+ "execution_count": 92,
"metadata": {},
"output_type": "execute_result"
}
@@ -2481,7 +2472,7 @@
},
{
"cell_type": "code",
- "execution_count": 143,
+ "execution_count": 93,
"metadata": {},
"outputs": [
{
@@ -2680,7 +2671,7 @@
"<Dot visualization: expr_as_graph [(\"K5\",(1..5)*(1..5))]>"
]
},
- "execution_count": 143,
+ "execution_count": 93,
"metadata": {},
"output_type": "execute_result"
}
@@ -2691,7 +2682,7 @@
},
{
"cell_type": "code",
- "execution_count": 136,
+ "execution_count": 94,
"metadata": {},
"outputs": [
{
@@ -2700,7 +2691,7 @@
"Preference changed: DOT_ENGINE = circo\n"
]
},
- "execution_count": 136,
+ "execution_count": 94,
"metadata": {},
"output_type": "execute_result"
}
@@ -2711,7 +2702,7 @@
},
{
"cell_type": "code",
- "execution_count": 141,
+ "execution_count": 95,
"metadata": {},
"outputs": [
{
@@ -3385,7 +3376,7 @@
"<Dot visualization: expr_as_graph [(\"K10\",(1..10)*(1..10))]>"
]
},
- "execution_count": 141,
+ "execution_count": 95,
"metadata": {},
"output_type": "execute_result"
}
@@ -3396,7 +3387,7 @@
},
{
"cell_type": "code",
- "execution_count": 140,
+ "execution_count": 96,
"metadata": {},
"outputs": [
{
@@ -5920,7 +5911,7 @@
"<Dot visualization: expr_as_graph [(\"K20\",(1..20)*(1..20))]>"
]
},
- "execution_count": 140,
+ "execution_count": 96,
"metadata": {},
"output_type": "execute_result"
}
@@ -5931,7 +5922,7 @@
},
{
"cell_type": "code",
- "execution_count": 144,
+ "execution_count": 97,
"metadata": {},
"outputs": [
{
@@ -5940,7 +5931,7 @@
"Preference changed: DOT_ENGINE = dot\n"
]
},
- "execution_count": 144,
+ "execution_count": 97,
"metadata": {},
"output_type": "execute_result"
}
@@ -5965,7 +5956,7 @@
},
{
"cell_type": "code",
- "execution_count": 164,
+ "execution_count": 98,
"metadata": {},
"outputs": [
{
@@ -5977,7 +5968,7 @@
"{(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)}"
]
},
- "execution_count": 164,
+ "execution_count": 98,
"metadata": {},
"output_type": "execute_result"
}
@@ -5988,7 +5979,7 @@
},
{
"cell_type": "code",
- "execution_count": 165,
+ "execution_count": 99,
"metadata": {},
"outputs": [
{
@@ -6000,7 +5991,7 @@
"{2,4,6,8,10}"
]
},
- "execution_count": 165,
+ "execution_count": 99,
"metadata": {},
"output_type": "execute_result"
}
@@ -6011,7 +6002,7 @@
},
{
"cell_type": "code",
- "execution_count": 166,
+ "execution_count": 100,
"metadata": {},
"outputs": [
{
@@ -6023,7 +6014,7 @@
"{1,2,3,4,5}"
]
},
- "execution_count": 166,
+ "execution_count": 100,
"metadata": {},
"output_type": "execute_result"
}
@@ -6034,7 +6025,7 @@
},
{
"cell_type": "code",
- "execution_count": 167,
+ "execution_count": 103,
"metadata": {},
"outputs": [
{
@@ -6046,7 +6037,7 @@
"{2,4,6,8,10}"
]
},
- "execution_count": 167,
+ "execution_count": 103,
"metadata": {},
"output_type": "execute_result"
}
@@ -6057,30 +6048,16 @@
},
{
"cell_type": "code",
- "execution_count": 73,
- "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": 73,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- ":let h {a,b| a∈1..10 ∧ b∈1..10 & b*2=a}"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 15,
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ ":let h {a,b| a∈1..10 ∧ b∈1..10 & b*2=a}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 111,
"metadata": {},
"outputs": [
{
@@ -6174,7 +6151,7 @@
"<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)}(\"h\",h)]>"
]
},
- "execution_count": 15,
+ "execution_count": 111,
"metadata": {},
"output_type": "execute_result"
}
@@ -6185,7 +6162,7 @@
},
{
"cell_type": "code",
- "execution_count": 9,
+ "execution_count": 104,
"metadata": {},
"outputs": [
{
@@ -6197,7 +6174,7 @@
"{(1↦\"nein\"),(2↦\"nein\"),(3↦\"ja\"),(4↦\"nein\"),(5↦\"nein\"),(6↦\"ja\"),(7↦\"nein\")}"
]
},
- "execution_count": 9,
+ "execution_count": 104,
"metadata": {},
"output_type": "execute_result"
}
@@ -6208,7 +6185,7 @@
},
{
"cell_type": "code",
- "execution_count": 10,
+ "execution_count": 105,
"metadata": {},
"outputs": [
{
@@ -6220,7 +6197,7 @@
"{1,2,3,4,5,6,7}"
]
},
- "execution_count": 10,
+ "execution_count": 105,
"metadata": {},
"output_type": "execute_result"
}
@@ -6231,7 +6208,7 @@
},
{
"cell_type": "code",
- "execution_count": 11,
+ "execution_count": 106,
"metadata": {},
"outputs": [
{
@@ -6243,7 +6220,7 @@
"{\"ja\",\"nein\"}"
]
},
- "execution_count": 11,
+ "execution_count": 106,
"metadata": {},
"output_type": "execute_result"
}
@@ -6254,7 +6231,7 @@
},
{
"cell_type": "code",
- "execution_count": 12,
+ "execution_count": 107,
"metadata": {},
"outputs": [
{
@@ -6362,10 +6339,10 @@
"</svg>"
],
"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)}(\"d\",d)]>"
+ "<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": 12,
+ "execution_count": 107,
"metadata": {},
"output_type": "execute_result"
}
@@ -6384,14 +6361,14 @@
" * $r[A] = \\{b \\mid \\exists a.((a,b)\\in r \\wedge a\\in A)\\}$\n",
"\n",
"Umkehrrelation:\n",
- " * $r^{-1} = \\{(b,a) \\mid (a,b)\\in r \\}$\\\\\n",
+ " * $r^{-1} = \\{(b,a) \\mid (a,b)\\in r \\}$\n",
"\n",
"Im Notebook muss für die Umkehrrelation leider Tilde (~) verwendet werden: "
]
},
{
"cell_type": "code",
- "execution_count": 16,
+ "execution_count": 112,
"metadata": {},
"outputs": [
{
@@ -6403,7 +6380,7 @@
"{4,5}"
]
},
- "execution_count": 16,
+ "execution_count": 112,
"metadata": {},
"output_type": "execute_result"
}
@@ -6414,7 +6391,7 @@
},
{
"cell_type": "code",
- "execution_count": 17,
+ "execution_count": 110,
"metadata": {},
"outputs": [
{
@@ -6426,7 +6403,7 @@
"{\"ja\"}"
]
},
- "execution_count": 17,
+ "execution_count": 110,
"metadata": {},
"output_type": "execute_result"
}
@@ -6437,30 +6414,16 @@
},
{
"cell_type": "code",
- "execution_count": 19,
+ "execution_count": null,
"metadata": {},
- "outputs": [
- {
- "data": {
- "text/markdown": [
- "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$"
- ],
- "text/plain": [
- "∅"
- ]
- },
- "execution_count": 19,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
+ "outputs": [],
"source": [
"d[{0}]"
]
},
{
"cell_type": "code",
- "execution_count": 26,
+ "execution_count": 113,
"metadata": {},
"outputs": [
{
@@ -6472,7 +6435,7 @@
"{(\"ja\"↦3),(\"ja\"↦6),(\"nein\"↦1),(\"nein\"↦2),(\"nein\"↦4),(\"nein\"↦5),(\"nein\"↦7)}"
]
},
- "execution_count": 26,
+ "execution_count": 113,
"metadata": {},
"output_type": "execute_result"
}
@@ -6483,7 +6446,7 @@
},
{
"cell_type": "code",
- "execution_count": 38,
+ "execution_count": 114,
"metadata": {},
"outputs": [
{
@@ -6594,7 +6557,7 @@
"<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": 38,
+ "execution_count": 114,
"metadata": {},
"output_type": "execute_result"
}
@@ -6607,12 +6570,12 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Man kann natürlich auch beide Operatore verknüpfen, zum Beispiel um herauszufinden welche Zahlen durch 3 teilbar sind:"
+ "Man kann natürlich auch beide Operatoren verknüpfen, zum Beispiel um herauszufinden welche Zahlen durch 3 teilbar sind:"
]
},
{
"cell_type": "code",
- "execution_count": 40,
+ "execution_count": 118,
"metadata": {},
"outputs": [
{
@@ -6624,7 +6587,7 @@
"{3,6}"
]
},
- "execution_count": 40,
+ "execution_count": 118,
"metadata": {},
"output_type": "execute_result"
}
@@ -6635,7 +6598,7 @@
},
{
"cell_type": "code",
- "execution_count": 54,
+ "execution_count": 119,
"metadata": {},
"outputs": [],
"source": [
@@ -6643,24 +6606,8 @@
]
},
{
- "cell_type": "code",
- "execution_count": 32,
+ "cell_type": "markdown",
"metadata": {},
- "outputs": [
- {
- "data": {
- "text/latex": [
- "$1 = 2$"
- ],
- "text/plain": [
- "1 = 2"
- ]
- },
- "execution_count": 32,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
"source": [
"# Binäre Relationen - Verknüpfung\n",
" \n",
@@ -6671,7 +6618,7 @@
},
{
"cell_type": "code",
- "execution_count": 42,
+ "execution_count": 121,
"metadata": {},
"outputs": [
{
@@ -6683,7 +6630,7 @@
"{(1↦4),(1↦8)}"
]
},
- "execution_count": 42,
+ "execution_count": 121,
"metadata": {},
"output_type": "execute_result"
}
@@ -6703,7 +6650,7 @@
},
{
"cell_type": "code",
- "execution_count": 49,
+ "execution_count": 123,
"metadata": {},
"outputs": [
{
@@ -6715,7 +6662,7 @@
"{(4↦1),(8↦2)}"
]
},
- "execution_count": 49,
+ "execution_count": 123,
"metadata": {},
"output_type": "execute_result"
}
@@ -6726,221 +6673,7 @@
},
{
"cell_type": "code",
- "execution_count": 50,
- "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 2.28.0 (20110509.1545)\n",
- " -->\n",
- "<!-- Title: state Pages: 1 -->\n",
- "<svg width=\"910pt\" height=\"256pt\"\n",
- " viewBox=\"0.00 0.00 910.00 256.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
- "<g id=\"graph1\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 252)\">\n",
- "<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"white\" points=\"-4,5 -4,-252 907,-252 907,5 -4,5\"/>\n",
- "<!-- abs2_3 -->\n",
- "<g id=\"node1\" class=\"node\"><title>abs2_3</title>\n",
- "<ellipse fill=\"none\" stroke=\"black\" cx=\"114\" cy=\"-222\" rx=\"38.0899\" ry=\"18\"/>\n",
- "<text text-anchor=\"middle\" x=\"114\" y=\"-217.8\" font-family=\"Times,serif\" font-size=\"14.00\">abs2_3</text>\n",
- "</g>\n",
- "<!-- abs2_4 -->\n",
- "<g id=\"node3\" class=\"node\"><title>abs2_4</title>\n",
- "<ellipse fill=\"none\" stroke=\"black\" cx=\"456\" cy=\"-116\" rx=\"38.0899\" ry=\"18\"/>\n",
- "<text text-anchor=\"middle\" x=\"456\" y=\"-111.8\" font-family=\"Times,serif\" font-size=\"14.00\">abs2_4</text>\n",
- "</g>\n",
- "<!-- abs2_3->abs2_4 -->\n",
- "<g id=\"edge2\" class=\"edge\"><title>abs2_3->abs2_4</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M127.315,-205.029C141.732,-187.917 163.375,-162.817 168.598,-160 209.069,-138.172 337.623,-125.628 408.301,-120.213\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"408.581,-123.702 418.292,-119.466 408.059,-116.722 408.581,-123.702\"/>\n",
- "<text text-anchor=\"middle\" x=\"425.701\" y=\"-164.8\" font-family=\"Times,serif\" font-size=\"14.00\">LET h BE  h={(2|->1),(4|->2),(6|->3),(8|->4),(10|->5)} IN "h" |-> h |-> "hh" |-> (h ; h) END</text>\n",
- "</g>\n",
- "<!-- abs1_2 -->\n",
- "<g id=\"node5\" class=\"node\"><title>abs1_2</title>\n",
- "<ellipse fill=\"none\" stroke=\"black\" cx=\"38\" cy=\"-116\" rx=\"38.0899\" ry=\"18\"/>\n",
- "<text text-anchor=\"middle\" x=\"38\" y=\"-111.8\" font-family=\"Times,serif\" font-size=\"14.00\">abs1_2</text>\n",
- "</g>\n",
- "<!-- abs2_3->abs1_2 -->\n",
- "<g id=\"edge4\" class=\"edge\"><title>abs2_3->abs1_2</title>\n",
- "<path fill=\"none\" stroke=\"sienna\" d=\"M99.282,-205.132C92.1936,-197.211 83.7441,-187.345 76.7882,-178 68.358,-166.674 59.8465,-153.523 52.953,-142.337\"/>\n",
- "<polygon fill=\"sienna\" stroke=\"sienna\" points=\"55.7616,-140.219 47.5805,-133.487 49.7778,-143.851 55.7616,-140.219\"/>\n",
- "<text text-anchor=\"middle\" x=\"94.1059\" y=\"-164.8\" font-family=\"Times,serif\" font-size=\"14.00\">$from</text>\n",
- "</g>\n",
- "<!-- "hh" -->\n",
- "<g id=\"node7\" class=\"node\"><title>"hh"</title>\n",
- "<polygon fill=\"#fff68f\" stroke=\"#fff68f\" points=\"148,-134 94,-134 94,-98 148,-98 148,-134\"/>\n",
- "<text text-anchor=\"middle\" x=\"121\" y=\"-111.8\" font-family=\"Times,serif\" font-size=\"14.00\">"hh"</text>\n",
- "</g>\n",
- "<!-- abs2_3->"hh" -->\n",
- "<g id=\"edge6\" class=\"edge\"><title>abs2_3->"hh"</title>\n",
- "<path fill=\"none\" stroke=\"#473c8b\" d=\"M113.725,-203.883C113.663,-191.641 113.856,-174.784 115.004,-160 115.399,-154.913 116.005,-149.51 116.681,-144.314\"/>\n",
- "<polygon fill=\"#473c8b\" stroke=\"#473c8b\" points=\"120.19,-144.498 118.126,-134.106 113.259,-143.516 120.19,-144.498\"/>\n",
- "<text text-anchor=\"middle\" x=\"133.498\" y=\"-164.8\" font-family=\"Times,serif\" font-size=\"14.00\">$to_el</text>\n",
- "</g>\n",
- "<!-- "h" -->\n",
- "<g id=\"node9\" class=\"node\"><title>"h"</title>\n",
- "<polygon fill=\"#fff68f\" stroke=\"#fff68f\" points=\"65,-36 11,-36 11,-0 65,-0 65,-36\"/>\n",
- "<text text-anchor=\"middle\" x=\"38\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">"h"</text>\n",
- "</g>\n",
- "<!-- abs1_2->"h" -->\n",
- "<g id=\"edge8\" class=\"edge\"><title>abs1_2->"h"</title>\n",
- "<path fill=\"none\" stroke=\"sienna\" d=\"M38,-97.8371C38,-83.5031 38,-62.8066 38,-46.2158\"/>\n",
- "<polygon fill=\"sienna\" stroke=\"sienna\" points=\"41.5001,-46.0143 38,-36.0143 34.5001,-46.0143 41.5001,-46.0143\"/>\n",
- "<text text-anchor=\"middle\" x=\"55.1059\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">$from</text>\n",
- "</g>\n",
- "<!-- abs2_1 -->\n",
- "<g id=\"node11\" class=\"node\"><title>abs2_1</title>\n",
- "<ellipse fill=\"none\" stroke=\"black\" cx=\"671\" cy=\"-18\" rx=\"38.0899\" ry=\"18\"/>\n",
- "<text text-anchor=\"middle\" x=\"671\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">abs2_1</text>\n",
- "</g>\n",
- "<!-- abs1_2->abs2_1 -->\n",
- "<g id=\"edge10\" class=\"edge\"><title>abs1_2->abs2_1</title>\n",
- "<path fill=\"none\" stroke=\"#473c8b\" d=\"M60.2104,-101.03C67.8108,-96.843 76.5362,-92.6619 85,-90 277.376,-29.4962 520.258,-20.057 622.325,-18.9274\"/>\n",
- "<polygon fill=\"#473c8b\" stroke=\"#473c8b\" points=\"622.655,-22.4247 632.623,-18.8368 622.593,-15.425 622.655,-22.4247\"/>\n",
- "<text text-anchor=\"middle\" x=\"243.498\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">$to_el</text>\n",
- "</g>\n",
- "<!-- (10\\|-\\>5) -->\n",
- "<g id=\"node12\" class=\"node\"><title>(10\\|-\\>5)</title>\n",
- "<polygon fill=\"burlywood\" stroke=\"burlywood\" points=\"512.5,-91.2 512.5,-140.8 581.5,-140.8 581.5,-91.2 512.5,-91.2\"/>\n",
- "<text text-anchor=\"middle\" x=\"522.25\" y=\"-111.8\" font-family=\"Times,serif\" font-size=\"14.00\"> </text>\n",
- "<polyline fill=\"none\" stroke=\"burlywood\" points=\"532,-91.2 532,-140.8 \"/>\n",
- "<text text-anchor=\"middle\" x=\"547\" y=\"-124.2\" font-family=\"Times,serif\" font-size=\"14.00\">10</text>\n",
- "<polyline fill=\"none\" stroke=\"burlywood\" points=\"532,-116 562,-116 \"/>\n",
- "<text text-anchor=\"middle\" x=\"547\" y=\"-99.4\" font-family=\"Times,serif\" font-size=\"14.00\">5</text>\n",
- "<polyline fill=\"none\" stroke=\"burlywood\" points=\"562,-91.2 562,-140.8 \"/>\n",
- "<text text-anchor=\"middle\" x=\"571.75\" y=\"-111.8\" font-family=\"Times,serif\" font-size=\"14.00\"> </text>\n",
- "</g>\n",
- "<!-- (10\\|-\\>5)->abs2_1 -->\n",
- "<g id=\"edge12\" class=\"edge\"><title>(10\\|-\\>5)->abs2_1</title>\n",
- "<path fill=\"none\" stroke=\"#473c8b\" d=\"M567.931,-91.1053C579.116,-79.1378 593.536,-64.9134 608.004,-54 616.709,-47.434 626.788,-41.2823 636.296,-36.0239\"/>\n",
- "<polygon fill=\"#473c8b\" stroke=\"#473c8b\" points=\"638.009,-39.0766 645.174,-31.2715 634.706,-32.9052 638.009,-39.0766\"/>\n",
- "<text text-anchor=\"middle\" x=\"625.498\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">$to_el</text>\n",
- "</g>\n",
- "<!-- (8\\|-\\>4) -->\n",
- "<g id=\"node14\" class=\"node\"><title>(8\\|-\\>4)</title>\n",
- "<polygon fill=\"burlywood\" stroke=\"burlywood\" points=\"600,-91.2 600,-140.8 662,-140.8 662,-91.2 600,-91.2\"/>\n",
- "<text text-anchor=\"middle\" x=\"609.75\" y=\"-111.8\" font-family=\"Times,serif\" font-size=\"14.00\"> </text>\n",
- "<polyline fill=\"none\" stroke=\"burlywood\" points=\"619.5,-91.2 619.5,-140.8 \"/>\n",
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- "<polyline fill=\"none\" stroke=\"burlywood\" points=\"619.5,-116 642.5,-116 \"/>\n",
- "<text text-anchor=\"middle\" x=\"631\" y=\"-99.4\" font-family=\"Times,serif\" font-size=\"14.00\">4</text>\n",
- "<polyline fill=\"none\" stroke=\"burlywood\" points=\"642.5,-91.2 642.5,-140.8 \"/>\n",
- "<text text-anchor=\"middle\" x=\"652.25\" y=\"-111.8\" font-family=\"Times,serif\" font-size=\"14.00\"> </text>\n",
- "</g>\n",
- "<!-- (8\\|-\\>4)->abs2_1 -->\n",
- "<g id=\"edge14\" class=\"edge\"><title>(8\\|-\\>4)->abs2_1</title>\n",
- "<path fill=\"none\" stroke=\"#473c8b\" d=\"M639.459,-91.0441C643.605,-79.7155 648.815,-66.0602 654.004,-54 655.312,-50.9596 656.743,-47.8008 658.2,-44.6804\"/>\n",
- "<polygon fill=\"#473c8b\" stroke=\"#473c8b\" points=\"661.391,-46.121 662.559,-35.5908 655.079,-43.0939 661.391,-46.121\"/>\n",
- "<text text-anchor=\"middle\" x=\"671.498\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">$to_el</text>\n",
- "</g>\n",
- "<!-- (6\\|-\\>3) -->\n",
- "<g id=\"node16\" class=\"node\"><title>(6\\|-\\>3)</title>\n",
- "<polygon fill=\"burlywood\" stroke=\"burlywood\" points=\"680,-91.2 680,-140.8 742,-140.8 742,-91.2 680,-91.2\"/>\n",
- "<text text-anchor=\"middle\" x=\"689.75\" y=\"-111.8\" font-family=\"Times,serif\" font-size=\"14.00\"> </text>\n",
- "<polyline fill=\"none\" stroke=\"burlywood\" points=\"699.5,-91.2 699.5,-140.8 \"/>\n",
- "<text text-anchor=\"middle\" x=\"711\" y=\"-124.2\" font-family=\"Times,serif\" font-size=\"14.00\">6</text>\n",
- "<polyline fill=\"none\" stroke=\"burlywood\" points=\"699.5,-116 722.5,-116 \"/>\n",
- "<text text-anchor=\"middle\" x=\"711\" y=\"-99.4\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
- "<polyline fill=\"none\" stroke=\"burlywood\" points=\"722.5,-91.2 722.5,-140.8 \"/>\n",
- "<text text-anchor=\"middle\" x=\"732.25\" y=\"-111.8\" font-family=\"Times,serif\" font-size=\"14.00\"> </text>\n",
- "</g>\n",
- "<!-- (6\\|-\\>3)->abs2_1 -->\n",
- "<g id=\"edge16\" class=\"edge\"><title>(6\\|-\\>3)->abs2_1</title>\n",
- "<path fill=\"none\" stroke=\"#473c8b\" d=\"M703.134,-90.9667C699.218,-79.6211 694.212,-65.9739 689,-54 687.664,-50.9301 686.179,-47.7532 684.653,-44.6226\"/>\n",
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- "<g id=\"node18\" class=\"node\"><title>(4\\|-\\>2)</title>\n",
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- "<g id=\"edge18\" class=\"edge\"><title>(4\\|-\\>2)->abs2_1</title>\n",
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- "<!-- (2\\|-\\>1)->abs2_1 -->\n",
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- "execution_count": 48,
+ "execution_count": 124,
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"cell_type": "markdown",
"metadata": {},
"source": [
- "# Transitive und Reflexive Hülle\n",
+ "# Transitive Hülle\n",
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+ "execution_count": 125,
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- "execution_count": 81,
+ "execution_count": 128,
"metadata": {},
"outputs": [
{
@@ -7213,7 +6909,7 @@
"{(2↦1),(4↦1),(4↦2),(6↦3),(8↦1),(8↦2),(8↦4),(10↦5)}"
]
},
- "execution_count": 81,
+ "execution_count": 128,
"metadata": {},
"output_type": "execute_result"
}
@@ -7224,7 +6920,7 @@
},
{
"cell_type": "code",
- "execution_count": 79,
+ "execution_count": 129,
"metadata": {},
"outputs": [
{
@@ -7333,10 +7029,10 @@
"</svg>"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [ArhA={1,2,3} & r={(1,2),(2,3)} & h={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"h+\",closure1(h))]>"
+ "<Dot visualization: expr_as_graph [hh={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"h+\",closure1(h))]>"
]
},
- "execution_count": 79,
+ "execution_count": 129,
"metadata": {},
"output_type": "execute_result"
}
@@ -7347,7 +7043,7 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 130,
"metadata": {},
"outputs": [],
"source": [
@@ -7358,12 +7054,12 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Ein weiteres Beispiel ist folgende abgeänderte strikte Untermengenrelation $\\subset$ für $\\pow(1 .. 3) $:"
+ "Ein weiteres Beispiel ist folgende abgeänderte strikte Untermengenrelation $\\subset$ für $ℙ(1 .. 3) $:"
]
},
{
"cell_type": "code",
- "execution_count": 83,
+ "execution_count": 131,
"metadata": {},
"outputs": [
{
@@ -7375,18 +7071,18 @@
"{(∅↦{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": 83,
+ "execution_count": 131,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
- ":let sub1 {x,y|y:POW(1..3) & x <<:y & card(x)+1=card(y)}"
+ ":let sub1 {x,y|y:ℙ(1..3) & x ⊂ y & card(x)+1=card(y)}"
]
},
{
"cell_type": "code",
- "execution_count": 89,
+ "execution_count": 132,
"metadata": {},
"outputs": [
{
@@ -7395,7 +7091,7 @@
"Preference changed: DOT_DECOMPOSE_NODES = false\n"
]
},
- "execution_count": 89,
+ "execution_count": 132,
"metadata": {},
"output_type": "execute_result"
}
@@ -7406,7 +7102,7 @@
},
{
"cell_type": "code",
- "execution_count": 90,
+ "execution_count": 133,
"metadata": {},
"outputs": [
{
@@ -7539,10 +7235,10 @@
"</svg>"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [Arsub1hA={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})} & h={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"sub1\",sub1)]>"
+ "<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": 90,
+ "execution_count": 133,
"metadata": {},
"output_type": "execute_result"
}
@@ -7553,7 +7249,7 @@
},
{
"cell_type": "code",
- "execution_count": 92,
+ "execution_count": 134,
"metadata": {},
"outputs": [
{
@@ -7728,10 +7424,10 @@
"</svg>"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [Arsub1hA={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})} & h={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"sub1\",closure1(sub1))]>"
+ "<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": 92,
+ "execution_count": 134,
"metadata": {},
"output_type": "execute_result"
}
@@ -7755,7 +7451,7 @@
},
{
"cell_type": "code",
- "execution_count": 51,
+ "execution_count": 135,
"metadata": {},
"outputs": [
{
@@ -7767,7 +7463,7 @@
"{1,2,3}"
]
},
- "execution_count": 51,
+ "execution_count": 135,
"metadata": {},
"output_type": "execute_result"
}
@@ -7778,7 +7474,7 @@
},
{
"cell_type": "code",
- "execution_count": 93,
+ "execution_count": 136,
"metadata": {},
"outputs": [
{
@@ -7790,7 +7486,7 @@
"{(1↦2),(2↦3)}"
]
},
- "execution_count": 93,
+ "execution_count": 136,
"metadata": {},
"output_type": "execute_result"
}
@@ -7801,7 +7497,7 @@
},
{
"cell_type": "code",
- "execution_count": 94,
+ "execution_count": 137,
"metadata": {},
"outputs": [
{
@@ -7849,10 +7545,10 @@
"</svg>"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [Arsub1hA={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})} & h={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"r\",r)]>"
+ "<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": 94,
+ "execution_count": 137,
"metadata": {},
"output_type": "execute_result"
}
@@ -7863,7 +7559,7 @@
},
{
"cell_type": "code",
- "execution_count": 95,
+ "execution_count": 138,
"metadata": {},
"outputs": [
{
@@ -7875,7 +7571,7 @@
"{(1↦3)}"
]
},
- "execution_count": 95,
+ "execution_count": 138,
"metadata": {},
"output_type": "execute_result"
}
@@ -7893,7 +7589,7 @@
},
{
"cell_type": "code",
- "execution_count": 96,
+ "execution_count": 139,
"metadata": {},
"outputs": [
{
@@ -7905,7 +7601,7 @@
"{(1↦1),(2↦2),(3↦3)}"
]
},
- "execution_count": 96,
+ "execution_count": 139,
"metadata": {},
"output_type": "execute_result"
}
@@ -7923,7 +7619,7 @@
},
{
"cell_type": "code",
- "execution_count": 60,
+ "execution_count": 140,
"metadata": {},
"outputs": [
{
@@ -7935,7 +7631,7 @@
"{(1↦1),(1↦2),(1↦3),(2↦2),(2↦3),(3↦3)}"
]
},
- "execution_count": 60,
+ "execution_count": 140,
"metadata": {},
"output_type": "execute_result"
}
@@ -7946,7 +7642,7 @@
},
{
"cell_type": "code",
- "execution_count": 61,
+ "execution_count": 141,
"metadata": {},
"outputs": [
{
@@ -7982,148 +7678,75 @@
"<!-- 2->3 -->\n",
"<g id=\"edge4\" class=\"edge\"><title>2->3</title>\n",
"<path fill=\"none\" stroke=\"firebrick\" d=\"M55.0851,-89.614C50.0673,-76.9977 43.1896,-59.7052 37.4963,-45.3907\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"40.7294,-44.0489 33.7814,-36.0504 34.225,-46.636 40.7294,-44.0489\"/>\n",
- "<text text-anchor=\"middle\" x=\"52.8303\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
- "</g>\n",
- "<!-- 2->2 -->\n",
- "<g id=\"edge6\" class=\"edge\"><title>2->2</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M89.2408,-116.242C99.0239,-116.419 107,-113.672 107,-108 107,-104.544 104.038,-102.173 99.5105,-100.889\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"99.5639,-97.3739 89.2408,-99.7581 98.7975,-104.332 99.5639,-97.3739\"/>\n",
- "<text text-anchor=\"middle\" x=\"112.83\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
- "</g>\n",
- "<!-- 1 -->\n",
- "<g id=\"node6\" class=\"node\"><title>1</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-216 0,-216 0,-180 54,-180 54,-216\"/>\n",
- "<text text-anchor=\"middle\" x=\"27\" y=\"-193.8\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
- "</g>\n",
- "<!-- 1->3 -->\n",
- "<g id=\"edge8\" class=\"edge\"><title>1->3</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M23.0751,-179.897C20.0025,-165.587 15.9833,-144.583 14.3394,-126 12.9295,-110.062 12.9295,-105.938 14.3394,-90 15.6366,-75.3365 18.4126,-59.1664 21.0356,-45.9664\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"24.4777,-46.6042 23.0751,-36.1026 17.6227,-45.1867 24.4777,-46.6042\"/>\n",
- "<text text-anchor=\"middle\" x=\"20.8303\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
- "</g>\n",
- "<!-- 1->2 -->\n",
- "<g id=\"edge10\" class=\"edge\"><title>1->2</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M33.9149,-179.614C38.9327,-166.998 45.8104,-149.705 51.5037,-135.391\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"54.775,-136.636 55.2186,-126.05 48.2706,-134.049 54.775,-136.636\"/>\n",
- "<text text-anchor=\"middle\" x=\"52.8303\" y=\"-148.8\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
- "</g>\n",
- "<!-- 1->1 -->\n",
- "<g id=\"edge12\" class=\"edge\"><title>1->1</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.2408,-206.242C64.0239,-206.419 72,-203.672 72,-198 72,-194.544 69.0382,-192.173 64.5105,-190.889\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"64.5639,-187.374 54.2408,-189.758 63.7975,-194.332 64.5639,-187.374\"/>\n",
- "<text text-anchor=\"middle\" x=\"77.8303\" y=\"-193.8\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
- "</g>\n",
- "</g>\n",
- "</svg>"
- ],
- "text/plain": [
- "<Dot visualization: expr_as_graph [ArhA={1,2,3} & r={(1,2),(2,3)} & h={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"r*\",id(A)\\/r\\/(r;r)\\/((r;r);r)\\/(((r;r);r);r))]>"
- ]
- },
- "execution_count": 61,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "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": 63,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/markdown": [
- "$\\{(1\\mapsto 2),(1\\mapsto 3),(2\\mapsto 3)\\}$"
- ],
- "text/plain": [
- "{(1↦2),(1↦3),(2↦3)}"
- ]
- },
- "execution_count": 63,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "closure1(r)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 65,
- "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 2.28.0 (20110509.1545)\n",
- " -->\n",
- "<!-- Title: state Pages: 1 -->\n",
- "<svg width=\"102pt\" height=\"224pt\"\n",
- " viewBox=\"0.00 0.00 102.00 224.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
- "<g id=\"graph1\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 220)\">\n",
- "<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"white\" points=\"-4,5 -4,-220 99,-220 99,5 -4,5\"/>\n",
- "<!-- 2 -->\n",
- "<g id=\"node1\" class=\"node\"><title>2</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-126 0,-126 0,-90 54,-90 54,-126\"/>\n",
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- "<text text-anchor=\"middle\" x=\"67\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"40.7294,-44.0489 33.7814,-36.0504 34.225,-46.636 40.7294,-44.0489\"/>\n",
+ "<text text-anchor=\"middle\" x=\"52.8303\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
"</g>\n",
- "<!-- 2->3 -->\n",
- "<g id=\"edge2\" class=\"edge\"><title>2->3</title>\n",
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- "<text text-anchor=\"middle\" x=\"57.2776\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">r+</text>\n",
+ "<!-- 2->2 -->\n",
+ "<g id=\"edge6\" class=\"edge\"><title>2->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M89.2408,-116.242C99.0239,-116.419 107,-113.672 107,-108 107,-104.544 104.038,-102.173 99.5105,-100.889\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"99.5639,-97.3739 89.2408,-99.7581 98.7975,-104.332 99.5639,-97.3739\"/>\n",
+ "<text text-anchor=\"middle\" x=\"112.83\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
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- "<text text-anchor=\"middle\" x=\"67\" y=\"-193.8\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
+ "<g id=\"node6\" class=\"node\"><title>1</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-216 0,-216 0,-180 54,-180 54,-216\"/>\n",
+ "<text text-anchor=\"middle\" x=\"27\" y=\"-193.8\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
+ "</g>\n",
+ "<!-- 1->3 -->\n",
+ "<g id=\"edge8\" class=\"edge\"><title>1->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M23.0751,-179.897C20.0025,-165.587 15.9833,-144.583 14.3394,-126 12.9295,-110.062 12.9295,-105.938 14.3394,-90 15.6366,-75.3365 18.4126,-59.1664 21.0356,-45.9664\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"24.4777,-46.6042 23.0751,-36.1026 17.6227,-45.1867 24.4777,-46.6042\"/>\n",
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"</g>\n",
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- "<text text-anchor=\"middle\" x=\"57.2776\" y=\"-148.8\" font-family=\"Times,serif\" font-size=\"14.00\">r+</text>\n",
+ "<g id=\"edge10\" class=\"edge\"><title>1->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M33.9149,-179.614C38.9327,-166.998 45.8104,-149.705 51.5037,-135.391\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"54.775,-136.636 55.2186,-126.05 48.2706,-134.049 54.775,-136.636\"/>\n",
+ "<text text-anchor=\"middle\" x=\"52.8303\" y=\"-148.8\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
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+ "<!-- 1->1 -->\n",
+ "<g id=\"edge12\" class=\"edge\"><title>1->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.2408,-206.242C64.0239,-206.419 72,-203.672 72,-198 72,-194.544 69.0382,-192.173 64.5105,-190.889\"/>\n",
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+ "<text text-anchor=\"middle\" x=\"77.8303\" y=\"-193.8\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
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"</g>\n",
"</svg>"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [ArhA={1,2,3} & r={(1,2),(2,3)} & h={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"r+\",closure1(r))]>"
+ "<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*\",id(A)\\/r\\/(r;r)\\/((r;r);r)\\/(((r;r);r);r))]>"
]
},
- "execution_count": 65,
+ "execution_count": 141,
"metadata": {},
"output_type": "execute_result"
}
],
+ "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": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "closure1(r)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
"source": [
":dot expr_as_graph (\"r+\",closure1(r))"
]
@@ -8137,7 +7760,7 @@
},
{
"cell_type": "code",
- "execution_count": 69,
+ "execution_count": 142,
"metadata": {},
"outputs": [
{
@@ -8149,7 +7772,7 @@
"{1,2}"
]
},
- "execution_count": 69,
+ "execution_count": 142,
"metadata": {},
"output_type": "execute_result"
}
@@ -8167,30 +7790,16 @@
},
{
"cell_type": "code",
- "execution_count": 72,
+ "execution_count": null,
"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\\_\\_}\\}$"
- ],
- "text/plain": [
- "/*@symbolic*/ {z_,z__∣z_ ↦ z__ ∈ {(1↦2),(1↦3),(2↦3)} ∨ z_ = z__}"
- ]
- },
- "execution_count": 72,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
+ "outputs": [],
"source": [
"closure(r)"
]
},
{
"cell_type": "code",
- "execution_count": 71,
+ "execution_count": 143,
"metadata": {},
"outputs": [
{
@@ -8202,7 +7811,7 @@
"{1,2,3}"
]
},
- "execution_count": 71,
+ "execution_count": 143,
"metadata": {},
"output_type": "execute_result"
}
@@ -8213,7 +7822,7 @@
},
{
"cell_type": "code",
- "execution_count": 159,
+ "execution_count": 144,
"metadata": {},
"outputs": [],
"source": [
@@ -8237,7 +7846,7 @@
},
{
"cell_type": "code",
- "execution_count": 101,
+ "execution_count": 145,
"metadata": {},
"outputs": [
{
@@ -8249,7 +7858,7 @@
"{(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
- "execution_count": 101,
+ "execution_count": 145,
"metadata": {},
"output_type": "execute_result"
}
@@ -8267,7 +7876,7 @@
},
{
"cell_type": "code",
- "execution_count": 100,
+ "execution_count": 146,
"metadata": {},
"outputs": [
{
@@ -8400,10 +8009,10 @@
"</svg>"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [Arsub1hA={1,2,3} & r={(1,2),(2,3)} & sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})} & h={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"sub1\",sub1)]>"
+ "<Dot visualization: expr_as_graph [Asub1A={1,2,3} & sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})}(\"sub1\",sub1)]>"
]
},
- "execution_count": 100,
+ "execution_count": 146,
"metadata": {},
"output_type": "execute_result"
}
@@ -8421,23 +8030,9 @@
},
{
"cell_type": "code",
- "execution_count": 102,
+ "execution_count": null,
"metadata": {},
- "outputs": [
- {
- "data": {
- "text/markdown": [
- "$\\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\\})\\}$"
- ],
- "text/plain": [
- "{(∅↦{1}),(∅↦{2}),(∅↦{3}),({1}↦{1,2}),({1}↦{1,3}),({1,2}↦{1,2,3}),({1,3}↦{1,2,3}),({2}↦{1,2}),({2}↦{2,3}),({2,3}↦{1,2,3}),({3}↦{1,3}),({3}↦{2,3})}"
- ]
- },
- "execution_count": 102,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
+ "outputs": [],
"source": [
"sub1"
]
@@ -8456,7 +8051,7 @@
},
{
"cell_type": "code",
- "execution_count": 104,
+ "execution_count": 147,
"metadata": {},
"outputs": [
{
@@ -8468,7 +8063,7 @@
"FALSE"
]
},
- "execution_count": 104,
+ "execution_count": 147,
"metadata": {},
"output_type": "execute_result"
}
@@ -8479,7 +8074,7 @@
},
{
"cell_type": "code",
- "execution_count": 114,
+ "execution_count": 148,
"metadata": {},
"outputs": [
{
@@ -8491,18 +8086,18 @@
"FALSE"
]
},
- "execution_count": 114,
+ "execution_count": 148,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
- "sub1 ∈ POW(1..3) --> POW(1..3)"
+ "sub1 ∈ ℙ(1..3) --> ℙ(1..3)"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 149,
"metadata": {},
"outputs": [],
"source": [
@@ -8511,7 +8106,7 @@
},
{
"cell_type": "code",
- "execution_count": 107,
+ "execution_count": 150,
"metadata": {},
"outputs": [
{
@@ -8523,7 +8118,7 @@
"{(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
- "execution_count": 107,
+ "execution_count": 150,
"metadata": {},
"output_type": "execute_result"
}
@@ -8534,19 +8129,16 @@
},
{
"cell_type": "code",
- "execution_count": 108,
+ "execution_count": 151,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
- "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\mathit{TRUE}$\n",
+ "$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{A} = \\{1,2,3\\}$\n",
- "* $\\mathit{r} = \\{(1\\mapsto 2),(2\\mapsto 3)\\}$\n",
- "* $\\mathit{sub1} = \\{(\\emptyset\\mapsto\\{1\\}),(\\emptyset\\mapsto\\{2\\}),(\\emptyset\\mapsto\\{3\\}),(\\{1\\}\\mapsto\\{1,2\\}),(\\{1\\}\\mapsto\\{1,3\\}),(\\{1,2\\}\\mapsto\\{1,2,3\\}),(\\{1,3\\}\\mapsto\\{1,2,3\\}),(\\{2\\}\\mapsto\\{1,2\\}),(\\{2\\}\\mapsto\\{2,3\\}),(\\{2,3\\}\\mapsto\\{1,2,3\\}),(\\{3\\}\\mapsto\\{1,3\\}),(\\{3\\}\\mapsto\\{2,3\\})\\}$\n",
- "* $\\mathit{h} = \\{(2\\mapsto 1),(4\\mapsto 2),(6\\mapsto 3),(8\\mapsto 4),(10\\mapsto 5)\\}$\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": [
@@ -8554,13 +8146,10 @@
"\n",
"Solution:\n",
"\tA = {1,2,3}\n",
- "\tr = {(1↦2),(2↦3)}\n",
- "\tsub1 = {(∅↦{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})}\n",
- "\th = {(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)}\n",
"\ti10 = {(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
- "execution_count": 108,
+ "execution_count": 151,
"metadata": {},
"output_type": "execute_result"
}
@@ -8571,19 +8160,16 @@
},
{
"cell_type": "code",
- "execution_count": 111,
+ "execution_count": 152,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
- "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\mathit{TRUE}$\n",
+ "$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{A} = \\{1,2,3\\}$\n",
- "* $\\mathit{r} = \\{(1\\mapsto 2),(2\\mapsto 3)\\}$\n",
- "* $\\mathit{sub1} = \\{(\\emptyset\\mapsto\\{1\\}),(\\emptyset\\mapsto\\{2\\}),(\\emptyset\\mapsto\\{3\\}),(\\{1\\}\\mapsto\\{1,2\\}),(\\{1\\}\\mapsto\\{1,3\\}),(\\{1,2\\}\\mapsto\\{1,2,3\\}),(\\{1,3\\}\\mapsto\\{1,2,3\\}),(\\{2\\}\\mapsto\\{1,2\\}),(\\{2\\}\\mapsto\\{2,3\\}),(\\{2,3\\}\\mapsto\\{1,2,3\\}),(\\{3\\}\\mapsto\\{1,3\\}),(\\{3\\}\\mapsto\\{2,3\\})\\}$\n",
- "* $\\mathit{h} = \\{(2\\mapsto 1),(4\\mapsto 2),(6\\mapsto 3),(8\\mapsto 4),(10\\mapsto 5)\\}$\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": [
@@ -8591,13 +8177,10 @@
"\n",
"Solution:\n",
"\tA = {1,2,3}\n",
- "\tr = {(1↦2),(2↦3)}\n",
- "\tsub1 = {(∅↦{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})}\n",
- "\th = {(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)}\n",
"\ti10 = {(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
- "execution_count": 111,
+ "execution_count": 152,
"metadata": {},
"output_type": "execute_result"
}
@@ -8615,19 +8198,16 @@
},
{
"cell_type": "code",
- "execution_count": 115,
+ "execution_count": 153,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
- "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\mathit{TRUE}$\n",
+ "$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{A} = \\{1,2,3\\}$\n",
- "* $\\mathit{r} = \\{(1\\mapsto 2),(2\\mapsto 3)\\}$\n",
- "* $\\mathit{sub1} = \\{(\\emptyset\\mapsto\\{1\\}),(\\emptyset\\mapsto\\{2\\}),(\\emptyset\\mapsto\\{3\\}),(\\{1\\}\\mapsto\\{1,2\\}),(\\{1\\}\\mapsto\\{1,3\\}),(\\{1,2\\}\\mapsto\\{1,2,3\\}),(\\{1,3\\}\\mapsto\\{1,2,3\\}),(\\{2\\}\\mapsto\\{1,2\\}),(\\{2\\}\\mapsto\\{2,3\\}),(\\{2,3\\}\\mapsto\\{1,2,3\\}),(\\{3\\}\\mapsto\\{1,3\\}),(\\{3\\}\\mapsto\\{2,3\\})\\}$\n",
- "* $\\mathit{h} = \\{(2\\mapsto 1),(4\\mapsto 2),(6\\mapsto 3),(8\\mapsto 4),(10\\mapsto 5)\\}$\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": [
@@ -8635,13 +8215,10 @@
"\n",
"Solution:\n",
"\tA = {1,2,3}\n",
- "\tr = {(1↦2),(2↦3)}\n",
- "\tsub1 = {(∅↦{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})}\n",
- "\th = {(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)}\n",
"\ti10 = {(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
]
},
- "execution_count": 115,
+ "execution_count": 153,
"metadata": {},
"output_type": "execute_result"
}
@@ -8659,7 +8236,7 @@
},
{
"cell_type": "code",
- "execution_count": 116,
+ "execution_count": 154,
"metadata": {},
"outputs": [
{
@@ -8671,7 +8248,7 @@
"FALSE"
]
},
- "execution_count": 116,
+ "execution_count": 154,
"metadata": {},
"output_type": "execute_result"
}
@@ -8682,7 +8259,7 @@
},
{
"cell_type": "code",
- "execution_count": 117,
+ "execution_count": 155,
"metadata": {},
"outputs": [
{
@@ -8694,7 +8271,7 @@
"FALSE"
]
},
- "execution_count": 117,
+ "execution_count": 155,
"metadata": {},
"output_type": "execute_result"
}
@@ -8703,6 +8280,29 @@
"i10 ∈ 0..10 --> 0..100"
]
},
+ {
+ "cell_type": "code",
+ "execution_count": 156,
+ "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": 156,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "dom(i10)"
+ ]
+ },
{
"cell_type": "markdown",
"metadata": {},
@@ -8712,19 +8312,16 @@
},
{
"cell_type": "code",
- "execution_count": 118,
+ "execution_count": 157,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
- "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\mathit{TRUE}$\n",
+ "$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
"* $\\mathit{A} = \\{1,2,3\\}$\n",
- "* $\\mathit{r} = \\{(1\\mapsto 2),(2\\mapsto 3)\\}$\n",
- "* $\\mathit{sub1} = \\{(\\emptyset\\mapsto\\{1\\}),(\\emptyset\\mapsto\\{2\\}),(\\emptyset\\mapsto\\{3\\}),(\\{1\\}\\mapsto\\{1,2\\}),(\\{1\\}\\mapsto\\{1,3\\}),(\\{1,2\\}\\mapsto\\{1,2,3\\}),(\\{1,3\\}\\mapsto\\{1,2,3\\}),(\\{2\\}\\mapsto\\{1,2\\}),(\\{2\\}\\mapsto\\{2,3\\}),(\\{2,3\\}\\mapsto\\{1,2,3\\}),(\\{3\\}\\mapsto\\{1,3\\}),(\\{3\\}\\mapsto\\{2,3\\})\\}$\n",
- "* $\\mathit{h} = \\{(2\\mapsto 1),(4\\mapsto 2),(6\\mapsto 3),(8\\mapsto 4),(10\\mapsto 5)\\}$\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",
"* $\\mathit{z} = 5$"
],
@@ -8733,14 +8330,11 @@
"\n",
"Solution:\n",
"\tA = {1,2,3}\n",
- "\tr = {(1↦2),(2↦3)}\n",
- "\tsub1 = {(∅↦{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})}\n",
- "\th = {(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)}\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": 118,
+ "execution_count": 157,
"metadata": {},
"output_type": "execute_result"
}
@@ -8751,7 +8345,7 @@
},
{
"cell_type": "code",
- "execution_count": 119,
+ "execution_count": 158,
"metadata": {},
"outputs": [
{
@@ -8763,7 +8357,7 @@
"{1,2,3,4,5,6,7,8,9,10}"
]
},
- "execution_count": 119,
+ "execution_count": 158,
"metadata": {},
"output_type": "execute_result"
}
@@ -8774,7 +8368,7 @@
},
{
"cell_type": "code",
- "execution_count": 120,
+ "execution_count": 159,
"metadata": {},
"outputs": [
{
@@ -8786,7 +8380,7 @@
"{2,3,4,5,6,7,8,9,10,11}"
]
},
- "execution_count": 120,
+ "execution_count": 159,
"metadata": {},
"output_type": "execute_result"
}
@@ -8797,7 +8391,7 @@
},
{
"cell_type": "code",
- "execution_count": 122,
+ "execution_count": 160,
"metadata": {},
"outputs": [
{
@@ -8809,7 +8403,7 @@
"{5,6}"
]
},
- "execution_count": 122,
+ "execution_count": 160,
"metadata": {},
"output_type": "execute_result"
}
@@ -8820,7 +8414,7 @@
},
{
"cell_type": "code",
- "execution_count": 123,
+ "execution_count": 167,
"metadata": {},
"outputs": [
{
@@ -8832,7 +8426,7 @@
"{5}"
]
},
- "execution_count": 123,
+ "execution_count": 167,
"metadata": {},
"output_type": "execute_result"
}
@@ -8843,7 +8437,7 @@
},
{
"cell_type": "code",
- "execution_count": 124,
+ "execution_count": 162,
"metadata": {},
"outputs": [
{
@@ -8855,7 +8449,7 @@
"{3}"
]
},
- "execution_count": 124,
+ "execution_count": 162,
"metadata": {},
"output_type": "execute_result"
}
@@ -8866,7 +8460,7 @@
},
{
"cell_type": "code",
- "execution_count": 125,
+ "execution_count": 163,
"metadata": {},
"outputs": [
{
@@ -8878,7 +8472,7 @@
"{6}"
]
},
- "execution_count": 125,
+ "execution_count": 163,
"metadata": {},
"output_type": "execute_result"
}
@@ -8889,7 +8483,7 @@
},
{
"cell_type": "code",
- "execution_count": 126,
+ "execution_count": 164,
"metadata": {},
"outputs": [
{
@@ -8901,7 +8495,7 @@
"{5,6,7,8,9,10,11}"
]
},
- "execution_count": 126,
+ "execution_count": 164,
"metadata": {},
"output_type": "execute_result"
}
@@ -8912,7 +8506,7 @@
},
{
"cell_type": "code",
- "execution_count": 127,
+ "execution_count": 165,
"metadata": {},
"outputs": [
{
@@ -8924,7 +8518,7 @@
"{1,2,3}"
]
},
- "execution_count": 127,
+ "execution_count": 165,
"metadata": {},
"output_type": "execute_result"
}
@@ -8942,7 +8536,7 @@
},
{
"cell_type": "code",
- "execution_count": 128,
+ "execution_count": 170,
"metadata": {},
"outputs": [
{
@@ -8954,7 +8548,7 @@
"5"
]
},
- "execution_count": 128,
+ "execution_count": 170,
"metadata": {},
"output_type": "execute_result"
}
@@ -8965,7 +8559,7 @@
},
{
"cell_type": "code",
- "execution_count": 129,
+ "execution_count": 171,
"metadata": {},
"outputs": [
{
@@ -8977,7 +8571,7 @@
"10"
]
},
- "execution_count": 129,
+ "execution_count": 171,
"metadata": {},
"output_type": "execute_result"
}
@@ -8995,7 +8589,7 @@
},
{
"cell_type": "code",
- "execution_count": 130,
+ "execution_count": 172,
"metadata": {},
"outputs": [
{
@@ -9007,7 +8601,7 @@
"∅"
]
},
- "execution_count": 130,
+ "execution_count": 172,
"metadata": {},
"output_type": "execute_result"
}
@@ -9018,17 +8612,40 @@
},
{
"cell_type": "code",
- "execution_count": 131,
+ "execution_count": 173,
+ "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": 173,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "i10"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 174,
"metadata": {},
"outputs": [
{
"ename": "CommandExecutionException",
- "evalue": ":eval: NOT-WELL-DEFINED: \nfunction applied outside of domain (#7): Function argument: 100, function value: {}\n ### Line: 1, Column: 316 until 324\n\n",
+ "evalue": ":eval: NOT-WELL-DEFINED: \nfunction applied outside of domain (#7): Function argument: 100, function value: {}\n ### Line: 1, Column: 99 until 107\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: 1, Column: 316 until 324\u001b[0m"
+ "\u001b[1m\u001b[31m ### Line: 1, Column: 99 until 107\u001b[0m"
]
}
],
@@ -9045,7 +8662,7 @@
},
{
"cell_type": "code",
- "execution_count": 135,
+ "execution_count": 175,
"metadata": {},
"outputs": [
{
@@ -9057,7 +8674,7 @@
"{{(1↦1),(2↦1)},{(1↦1),(2↦2)},{(1↦2),(2↦1)},{(1↦2),(2↦2)}}"
]
},
- "execution_count": 135,
+ "execution_count": 175,
"metadata": {},
"output_type": "execute_result"
}
@@ -9081,7 +8698,7 @@
},
{
"cell_type": "code",
- "execution_count": 137,
+ "execution_count": 176,
"metadata": {},
"outputs": [
{
@@ -9093,7 +8710,7 @@
"{∅,{(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": 137,
+ "execution_count": 176,
"metadata": {},
"output_type": "execute_result"
}
@@ -9102,6 +8719,37 @@
"1..2 +-> 1..2"
]
},
+ {
+ "cell_type": "code",
+ "execution_count": 177,
+ "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": 177,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "(1..2 --> 1..2) <: (1..2 +-> 1..2)"
+ ]
+ },
{
"cell_type": "markdown",
"metadata": {},
@@ -9111,7 +8759,7 @@
},
{
"cell_type": "code",
- "execution_count": 141,
+ "execution_count": 178,
"metadata": {},
"outputs": [
{
@@ -9123,7 +8771,7 @@
"{∅,{(1↦1)},{(1↦2)},{(2↦1)},{(2↦2)}}"
]
},
- "execution_count": 141,
+ "execution_count": 178,
"metadata": {},
"output_type": "execute_result"
}
@@ -9141,7 +8789,7 @@
},
{
"cell_type": "code",
- "execution_count": 143,
+ "execution_count": 179,
"metadata": {},
"outputs": [
{
@@ -9153,7 +8801,7 @@
"{{(1↦1),(2↦2)},{(1↦2),(2↦1)}}"
]
},
- "execution_count": 143,
+ "execution_count": 179,
"metadata": {},
"output_type": "execute_result"
}
@@ -9171,7 +8819,7 @@
},
{
"cell_type": "code",
- "execution_count": 139,
+ "execution_count": 180,
"metadata": {},
"outputs": [
{
@@ -9183,7 +8831,7 @@
"{{(1↦1),(2↦2)},{(1↦2),(2↦1)}}"
]
},
- "execution_count": 139,
+ "execution_count": 180,
"metadata": {},
"output_type": "execute_result"
}
@@ -9194,7 +8842,7 @@
},
{
"cell_type": "code",
- "execution_count": 140,
+ "execution_count": 181,
"metadata": {},
"outputs": [
{
@@ -9206,7 +8854,7 @@
"{{(1↦1),(2↦2)},{(1↦2),(2↦1)}}"
]
},
- "execution_count": 140,
+ "execution_count": 181,
"metadata": {},
"output_type": "execute_result"
}
@@ -9217,7 +8865,7 @@
},
{
"cell_type": "code",
- "execution_count": 160,
+ "execution_count": 182,
"metadata": {},
"outputs": [],
"source": [
@@ -9230,7 +8878,7 @@
"source": [
"# Endliche Folgen\n",
" \n",
- "Es gibt verschiedene Schreibweisen für endliche Folgen. \\\\\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",
@@ -9268,7 +8916,7 @@
},
{
"cell_type": "code",
- "execution_count": 146,
+ "execution_count": 199,
"metadata": {},
"outputs": [
{
@@ -9280,7 +8928,7 @@
"{(1↦22),(2↦22),(3↦33)}"
]
},
- "execution_count": 146,
+ "execution_count": 199,
"metadata": {},
"output_type": "execute_result"
}
@@ -9291,7 +8939,7 @@
},
{
"cell_type": "code",
- "execution_count": 147,
+ "execution_count": 200,
"metadata": {},
"outputs": [
{
@@ -9303,7 +8951,7 @@
"{(1↦22),(2↦22),(3↦33)}"
]
},
- "execution_count": 147,
+ "execution_count": 200,
"metadata": {},
"output_type": "execute_result"
}
@@ -9314,7 +8962,7 @@
},
{
"cell_type": "code",
- "execution_count": 158,
+ "execution_count": 190,
"metadata": {},
"outputs": [
{
@@ -9378,10 +9026,10 @@
"</svg>"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [Arsub1Ghi10A={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})} & G={(1,22),(2,22),(3,33)} & h={(2,1),(4,2),(6,3),(8,4),(10,5)} & i10={(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),(8,9),(9,10),(10,11)}(\"G\",G)]>"
+ "<Dot visualization: expr_as_graph [AGA={1,2,3} & G={(1,22),(2,22),(3,33)}(\"G\",G)]>"
]
},
- "execution_count": 158,
+ "execution_count": 190,
"metadata": {},
"output_type": "execute_result"
}
@@ -9405,7 +9053,7 @@
},
{
"cell_type": "code",
- "execution_count": 149,
+ "execution_count": 191,
"metadata": {},
"outputs": [
{
@@ -9417,7 +9065,7 @@
"22"
]
},
- "execution_count": 149,
+ "execution_count": 191,
"metadata": {},
"output_type": "execute_result"
}
@@ -9435,7 +9083,7 @@
},
{
"cell_type": "code",
- "execution_count": 150,
+ "execution_count": 192,
"metadata": {},
"outputs": [
{
@@ -9447,7 +9095,7 @@
"3"
]
},
- "execution_count": 150,
+ "execution_count": 192,
"metadata": {},
"output_type": "execute_result"
}
@@ -9458,7 +9106,7 @@
},
{
"cell_type": "code",
- "execution_count": 151,
+ "execution_count": 193,
"metadata": {},
"outputs": [
{
@@ -9470,7 +9118,7 @@
"3"
]
},
- "execution_count": 151,
+ "execution_count": 193,
"metadata": {},
"output_type": "execute_result"
}
@@ -9488,7 +9136,7 @@
},
{
"cell_type": "code",
- "execution_count": 153,
+ "execution_count": 194,
"metadata": {},
"outputs": [
{
@@ -9500,7 +9148,7 @@
"{1,2,3}"
]
},
- "execution_count": 153,
+ "execution_count": 194,
"metadata": {},
"output_type": "execute_result"
}
@@ -9511,7 +9159,7 @@
},
{
"cell_type": "code",
- "execution_count": 154,
+ "execution_count": 195,
"metadata": {},
"outputs": [
{
@@ -9523,7 +9171,7 @@
"{22,33}"
]
},
- "execution_count": 154,
+ "execution_count": 195,
"metadata": {},
"output_type": "execute_result"
}
@@ -9534,7 +9182,7 @@
},
{
"cell_type": "code",
- "execution_count": 155,
+ "execution_count": 196,
"metadata": {},
"outputs": [
{
@@ -9546,7 +9194,7 @@
"{1,2}"
]
},
- "execution_count": 155,
+ "execution_count": 196,
"metadata": {},
"output_type": "execute_result"
}
@@ -9557,7 +9205,7 @@
},
{
"cell_type": "code",
- "execution_count": 161,
+ "execution_count": 197,
"metadata": {},
"outputs": [],
"source": [