diff --git a/info4/kapitel-0/Mengentheorie.ipynb b/info4/kapitel-0/Mengentheorie.ipynb
index 3e5d2a5f07a46facb29de7f756ebfb9cf9919f31..8a2e8cc1d7b29a69f4a14496653871b424e599fc 100644
--- a/info4/kapitel-0/Mengentheorie.ipynb
+++ b/info4/kapitel-0/Mengentheorie.ipynb
@@ -741,6 +741,1129 @@
"Gesetze von De Morgan: $Op1( x ~ Op2 ~ y)$ wird umgewandelt nach $Op1(x) ~ Op2' ~ Op1(y)$, wo Op2' der duale Operator von Op2 ist."
]
},
+ {
+ "cell_type": "code",
+ "execution_count": 37,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$"
+ ],
+ "text/plain": [
+ "∅"
+ ]
+ },
+ "execution_count": 37,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{2,4} ∩ (ℤ \\ {2,4})"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Teilmenge\n",
+ "\n",
+ "\n",
+ "Eine Menge $x$ ist eine Teilmenge oder Untermenge von $y$ wenn gilt:\n",
+ "* $\\forall a. a\\in x \\Rightarrow a \\in y$\n",
+ "\n",
+ "Wir benutzen diese Schreibweise $x \\subseteq y$.\n",
+ "In diesem Fall ist $y$ auch eine Obermenge von $y$, geschrieben als $y \\supseteq x$.\n",
+ "\n",
+ "Für die echte Teilmenge benutzen wir folgende Schreibweise und Definition:\n",
+ "* $x \\subset y$ $\\Leftrightarrow$ $(x \\subseteq y \\wedge x\\neq y)$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 68,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\newcommand{\\upto}{\\mathbin{.\\mkern1mu.}}\\mathit{TRUE}$\n",
+ "\n",
+ "**Solution:**\n",
+ "* $\\mathit{x} = \\{1,3\\}$\n",
+ "* $\\mathit{y} = (1 \\upto 5)$"
+ ],
+ "text/plain": [
+ "TRUE\n",
+ "\n",
+ "Solution:\n",
+ "\tx = {1,3}\n",
+ "\ty = (1 ‥ 5)"
+ ]
+ },
+ "execution_count": 68,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "x={1,3} ∧ y = 1..5 ∧ ∀a.(a∈x ⇒ a∈y)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 67,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{FALSE}$"
+ ],
+ "text/plain": [
+ "FALSE"
+ ]
+ },
+ "execution_count": 67,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "x={1,3} ∧ y = 1..5 ∧ ∀a.(a∈y ⇒ a∈x)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 61,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{TRUE}$"
+ ],
+ "text/plain": [
+ "TRUE"
+ ]
+ },
+ "execution_count": 61,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{1,3} ⊂ 1..5"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Kardinalität\n",
+ "\n",
+ "Die Anzahl der Elemente einer Menge $x$ schreiben wir als \n",
+ "* $\\mid x\\mid$ oder auch als $card(x)$ (B Schreibweise)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 38,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$3$"
+ ],
+ "text/plain": [
+ "3"
+ ]
+ },
+ "execution_count": 38,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "card({1,2,3})"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 39,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$3$"
+ ],
+ "text/plain": [
+ "3"
+ ]
+ },
+ "execution_count": 39,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "card({1,1,2,3,2})"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 40,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$0$"
+ ],
+ "text/plain": [
+ "0"
+ ]
+ },
+ "execution_count": 40,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "card(∅)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Achtung, die Kardinalität kann auch unendlich sein: je nach Formalismus, ist folgender Ausdruck entweder unendlich oder nicht wohl definiert: $\\mid \\{x \\mid x>0\\} \\mid$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 41,
+ "metadata": {},
+ "outputs": [
+ {
+ "ename": "CommandExecutionException",
+ "evalue": ":eval: NOT-WELL-DEFINED: \ncard applied to very large set, cardinality not representable in ProB: closure([x],[integer],b(greater(b(identifier(...),integer,[...]),b(value(...),integer,[...])),pred,[nodeid(pos(...))]))\n\n",
+ "output_type": "error",
+ "traceback": [
+ "\u001b[1m\u001b[31m:eval: NOT-WELL-DEFINED: \u001b[0m",
+ "\u001b[1m\u001b[31mcard applied to very large set, cardinality not representable in ProB: closure([x],[integer],b(greater(b(identifier(...),integer,[...]),b(value(...),integer,[...])),pred,[nodeid(pos(...))]))\u001b[0m"
+ ]
+ }
+ ],
+ "source": [
+ "card({x|x>0})"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# SEND+MORE=MONEY\n",
+ "\n",
+ "Klassisches arithmetisches Puzzle wo acht unterschiedliche Ziffern gefunden werden sollen die folgende Gleichung erfüllen:\n",
+ "\n",
+ "| | | | | |\n",
+ "|---|---|---|---|---|\n",
+ "| | S | E | N | D |\n",
+ "| + | M | O | R | E |\n",
+ " |\n",
+ "|= M| O | N | E | Y |\n",
+ "| | | | | |\n",
+ "\n",
+ "Wir können dies nun in Logik, Mengentheorie und Arithmetik modellieren und lösen.\n",
+ " "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Wir haben acht Ziffern:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 69,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{TRUE}$\n",
+ "\n",
+ "**Solution:**\n",
+ "* $\\mathit{R} = 0$\n",
+ "* $\\mathit{S} = 0$\n",
+ "* $\\mathit{D} = 0$\n",
+ "* $\\mathit{E} = 0$\n",
+ "* $\\mathit{Y} = 0$\n",
+ "* $\\mathit{M} = 0$\n",
+ "* $\\mathit{N} = 0$\n",
+ "* $\\mathit{O} = 0$"
+ ],
+ "text/plain": [
+ "TRUE\n",
+ "\n",
+ "Solution:\n",
+ "\tR = 0\n",
+ "\tS = 0\n",
+ "\tD = 0\n",
+ "\tE = 0\n",
+ "\tY = 0\n",
+ "\tM = 0\n",
+ "\tN = 0\n",
+ "\tO = 0"
+ ]
+ },
+ "execution_count": 69,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{S,E,N,D,M,O,R,Y} ⊆ 0..9"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "diese Ziffern sind alle unterschiedlich:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 43,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{TRUE}$\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",
+ "* $\\mathit{O} = 2$"
+ ],
+ "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"
+ ]
+ },
+ "execution_count": 43,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{S,E,N,D,M,O,R,Y} ⊆ 0..9 ∧\n",
+ "card({S,E,N,D,M,O,R,Y}) = 8"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "und wobei die zwei führenden Ziffern S und M ungleich 0 sind:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 45,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{TRUE}$\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",
+ "* $\\mathit{O} = 2$"
+ ],
+ "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"
+ ]
+ },
+ "execution_count": 45,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{S,E,N,D,M,O,R,Y} ⊆ 0..9 ∧ card({S,E,N,D,M,O,R,Y}) = 8 ∧\n",
+ "S ≠ 0 & M ≠ 0"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "und wo die Summe von SEND+MORE MONEY ergibt:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 70,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{TRUE}$\n",
+ "\n",
+ "**Solution:**\n",
+ "* $\\mathit{R} = 8$\n",
+ "* $\\mathit{S} = 9$\n",
+ "* $\\mathit{D} = 7$\n",
+ "* $\\mathit{E} = 5$\n",
+ "* $\\mathit{Y} = 2$\n",
+ "* $\\mathit{M} = 1$\n",
+ "* $\\mathit{N} = 6$\n",
+ "* $\\mathit{O} = 0$"
+ ],
+ "text/plain": [
+ "TRUE\n",
+ "\n",
+ "Solution:\n",
+ "\tR = 8\n",
+ "\tS = 9\n",
+ "\tD = 7\n",
+ "\tE = 5\n",
+ "\tY = 2\n",
+ "\tM = 1\n",
+ "\tN = 6\n",
+ "\tO = 0"
+ ]
+ },
+ "execution_count": 70,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{S,E,N,D,M,O,R,Y} ⊆ 0..9 ∧ card({S,E,N,D,M,O,R,Y}) = 8 ∧ S ≠ 0 & M ≠ 0 ∧\n",
+ " S*1000 + E*100 + N*10 + D +\n",
+ " M*1000 + O*100 + R*10 + E =\n",
+ " M*10000 + O*1000 + N*100 + E*10 + Y"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Aber ist dies die einzige Lösung ? Wir können einfach die Menge aller Lösungen berechnen:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 48,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{(9\\mapsto 5\\mapsto 6\\mapsto 7\\mapsto 1\\mapsto 0\\mapsto 8\\mapsto 2)\\}$"
+ ],
+ "text/plain": [
+ "{(9↦5↦6↦7↦1↦0↦8↦2)}"
+ ]
+ },
+ "execution_count": 48,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ " {S,E,N,D, M,O,R, Y |\n",
+ " {S,E,N,D, M,O,R, Y} ⊆ 0..9 ∧ S >0 ∧ M >0 ∧ \n",
+ " card({S,E,N,D, M,O,R, Y}) = 8 ∧ \n",
+ " S*1000 + E*100 + N*10 + D +\n",
+ " M*1000 + O*100 + R*10 + E =\n",
+ " M*10000 + O*1000 + N*100 + E*10 + Y }"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 71,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "|S|E|N|D|M|O|R|Y|\n",
+ "|---|---|---|---|---|---|---|---|\n",
+ "|$9$|$5$|$6$|$7$|$1$|$0$|$8$|$2$|\n"
+ ],
+ "text/plain": [
+ "S\tE\tN\tD\tM\tO\tR\tY\n",
+ "9\t5\t6\t7\t1\t0\t8\t2\n"
+ ]
+ },
+ "execution_count": 71,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":table {S,E,N,D, M,O,R, Y |\n",
+ " {S,E,N,D, M,O,R, Y} ⊆ 0..9 ∧ S >0 ∧ M >0 ∧ \n",
+ " card({S,E,N,D, M,O,R, Y}) = 8 ∧ \n",
+ " S*1000 + E*100 + N*10 + D +\n",
+ " M*1000 + O*100 + R*10 + E =\n",
+ " M*10000 + O*1000 + N*100 + E*10 + Y }"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "| | | | | |\n",
+ "|---|---|---|---|---|\n",
+ "| | S=9 | E=5 | N=6 | D=7 |\n",
+ "| + | M=1 | O=0 | R=8 | E=5 |\n",
+ " |\n",
+ "|= M=1| O=0 | N=6 | E=5 | Y=2 |\n",
+ "| | | | | |"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Ein anderes arithmetisches Puzzle ist KISS*KISS=PASSION."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 72,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "|K|I|S|P|A|O|N|\n",
+ "|---|---|---|---|---|---|---|\n",
+ "|$2$|$0$|$3$|$4$|$1$|$8$|$9$|\n"
+ ],
+ "text/plain": [
+ "K\tI\tS\tP\tA\tO\tN\n",
+ "2\t0\t3\t4\t1\t8\t9\n"
+ ]
+ },
+ "execution_count": 72,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":table {K, I, S, P, A, O, N |\n",
+ " {K,P} ⊆ 1..9 ∧\n",
+ " {I,S,A,O,N} ⊆ 0..9 ∧\n",
+ " (1000*K+100*I+10*S+S) * (1000*K+100*I+10*S+S) \n",
+ " = 1000000*P+100000*A+10000*S+1000*S+100*I+10*O+N &\n",
+ " card({K, I, S, P, A, O, N}) = 7}"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Mengen von Mengen\n",
+ "\n",
+ "Mengen können selber auch Mengen beinhalten\n",
+ "Dies sind alles unterschiedliche Mengen:\n",
+ "* $\\{\\{2\\},\\{3,4\\}\\}$\n",
+ "* $\\{\\{2,3\\},\\{4\\}\\}$\n",
+ "* $\\{\\{2,3\\},\\{3,4\\}\\}$\n",
+ "* $\\{\\{2,3,4\\}\\}$\n",
+ "* $\\{2,3,4\\}$ \n",
+ "\n",
+ "Wir haben zum Beispiel:\n",
+ "* $\\mathit{card}(\\{\\{2,3,4\\}\\}) = 1$\n",
+ "* $\\mathit{card}(\\{\\{2\\},\\{3,4\\}\\}) = 2$\n",
+ "* $\\mathit{card}(\\{2,3,4\\}) = 3$\n",
+ "* $\\{2\\} \\in \\{\\{2\\},\\{3,4\\}\\} $\n",
+ "* $\\{2\\} \\not\\in \\{\\{2,3\\},\\{4\\}\\} $"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 57,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$3$"
+ ],
+ "text/plain": [
+ "3"
+ ]
+ },
+ "execution_count": 57,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "card({2,3,4})"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 52,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$1$"
+ ],
+ "text/plain": [
+ "1"
+ ]
+ },
+ "execution_count": 52,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "card({{2,3,4}})"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 53,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$2$"
+ ],
+ "text/plain": [
+ "2"
+ ]
+ },
+ "execution_count": 53,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "card({{2},{3,4}})"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 55,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{TRUE}$"
+ ],
+ "text/plain": [
+ "TRUE"
+ ]
+ },
+ "execution_count": 55,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{2} ∈ {{2}, {3,4}}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 56,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{FALSE}$"
+ ],
+ "text/plain": [
+ "FALSE"
+ ]
+ },
+ "execution_count": 56,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{2} ∈ {{2,3}, {4}}"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Mengen von Mengen und die leere Menge\n",
+ "\n",
+ "Achtung: man muss die leere Menge von der Menge die die leere Menge beinhaltet unterscheiden:\n",
+ "* $\\emptyset \\neq \\{\\emptyset\\}$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 76,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{FALSE}$"
+ ],
+ "text/plain": [
+ "FALSE"
+ ]
+ },
+ "execution_count": 76,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "∅ = {∅}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 77,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$0$"
+ ],
+ "text/plain": [
+ "0"
+ ]
+ },
+ "execution_count": 77,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "card(∅)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 78,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$1$"
+ ],
+ "text/plain": [
+ "1"
+ ]
+ },
+ "execution_count": 78,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "card({∅})"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 83,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{TRUE}$"
+ ],
+ "text/plain": [
+ "TRUE"
+ ]
+ },
+ "execution_count": 83,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "∅ ∈ {∅}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 84,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{FALSE}$"
+ ],
+ "text/plain": [
+ "FALSE"
+ ]
+ },
+ "execution_count": 84,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "∅ ∈ ∅"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 85,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\mathit{TRUE}$"
+ ],
+ "text/plain": [
+ "TRUE"
+ ]
+ },
+ "execution_count": 85,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "∅ ∉ ∅"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Potenzmenge\n",
+ "Die Menge aller Untermengen einer Menge $A$ schreiben wir als $\\pow(\\mathit{A}) $ oder auch als $2^{A}$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 86,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\{\\emptyset,\\{1\\},\\{1,2\\},\\{2\\}\\}$"
+ ],
+ "text/plain": [
+ "{∅,{1},{1,2},{2}}"
+ ]
+ },
+ "execution_count": 86,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "POW({1,2})"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 87,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\{\\emptyset\\}$"
+ ],
+ "text/plain": [
+ "{∅}"
+ ]
+ },
+ "execution_count": 87,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "POW(∅)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 88,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\{\\emptyset,\\{1\\},\\{1,2\\},\\{1,3\\},\\{2\\},\\{1,2,3\\},\\{2,3\\},\\{3\\}\\}$"
+ ],
+ "text/plain": [
+ "{∅,{1},{1,2},{1,3},{2},{1,2,3},{2,3},{3}}"
+ ]
+ },
+ "execution_count": 88,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "POW(1..3)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 98,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "|Elements|\n",
+ "|---|\n",
+ "|$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$|\n",
+ "|$\\{1\\}$|\n",
+ "|$\\{1,2\\}$|\n",
+ "|$\\{1,3\\}$|\n",
+ "|$\\{2\\}$|\n",
+ "|$\\{1,2,3\\}$|\n",
+ "|$\\{2,3\\}$|\n",
+ "|$\\{3\\}$|\n"
+ ],
+ "text/plain": [
+ "Elements\n",
+ "{}\n",
+ "{1}\n",
+ "{1,2}\n",
+ "{1,3}\n",
+ "{2}\n",
+ "{1,2,3}\n",
+ "{2,3}\n",
+ "{3}\n"
+ ]
+ },
+ "execution_count": 98,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":table POW(1..3)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 101,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "|Elements|\n",
+ "|---|\n",
+ "|$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$|\n",
+ "|$\\{\\emptyset\\}$|\n",
+ "|$\\{\\emptyset,\\{1\\}\\}$|\n",
+ "|$\\{\\{1\\}\\}$|\n"
+ ],
+ "text/plain": [
+ "Elements\n",
+ "{}\n",
+ "{{}}\n",
+ "{{},{1}}\n",
+ "{{1}}\n"
+ ]
+ },
+ "execution_count": 101,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":table POW(POW({1}))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 90,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$8$"
+ ],
+ "text/plain": [
+ "8"
+ ]
+ },
+ "execution_count": 90,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "card(POW(1..3))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 91,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$1024$"
+ ],
+ "text/plain": [
+ "1024"
+ ]
+ },
+ "execution_count": 91,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "card(POW(1..10))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 92,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$1267650600228229401496703205376$"
+ ],
+ "text/plain": [
+ "1267650600228229401496703205376"
+ ]
+ },
+ "execution_count": 92,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "card(POW(1..100))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 95,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$340282366920938463463374607431768211456$"
+ ],
+ "text/plain": [
+ "340282366920938463463374607431768211456"
+ ]
+ },
+ "execution_count": 95,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "card(POW(POW(1..7)))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 96,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$115792089237316195423570985008687907853269984665640564039457584007913129639936$"
+ ],
+ "text/plain": [
+ "115792089237316195423570985008687907853269984665640564039457584007913129639936"
+ ]
+ },
+ "execution_count": 96,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "card(POW(POW(POW(1..3))))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Relationen\n",
+ "* Was ist eine Relation?\n",
+ "* Wie kann man Relationen in Mengentheorie und Logik abbilden?"
+ ]
+ },
{
"cell_type": "code",
"execution_count": null,