diff --git a/info4/kapitel-0/Mengentheorie.ipynb b/info4/kapitel-0/Mengentheorie.ipynb
index e5bf56b464d4946cbd6368b73bf1de721558a28a..2ce3414ef4d19f763799e2b20e2f23df4aa292bc 100644
--- a/info4/kapitel-0/Mengentheorie.ipynb
+++ b/info4/kapitel-0/Mengentheorie.ipynb
@@ -5912,12 +5912,1650 @@
":pref DOT_ENGINE=dot"
]
},
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Definitions- und Wertebereich\n",
+ "\n",
+ "Definitionsbereich (Domain in Englisch):\n",
+ "* $dom(r) = \\{a \\mid \\exists b.((a,b)\\in r)\\}$\n",
+ "\n",
+ "Wertebereich (Bildmenge, Range in Englisch):\n",
+ "* $ran(r) = \\{b \\mid \\exists a.((a,b)\\in r)\\}$\n",
+ " "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 164,
+ "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": 164,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":let h {a,b| a∈1..10 ∧ b∈1..10 & b*2=a}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 165,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{2,4,6,8,10\\}$"
+ ],
+ "text/plain": [
+ "{2,4,6,8,10}"
+ ]
+ },
+ "execution_count": 165,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{a| ∃b.((a,b)∈h)}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 166,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{1,2,3,4,5\\}$"
+ ],
+ "text/plain": [
+ "{1,2,3,4,5}"
+ ]
+ },
+ "execution_count": 166,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "{b| ∃a.((a,b)∈h)}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 167,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{2,4,6,8,10\\}$"
+ ],
+ "text/plain": [
+ "{2,4,6,8,10}"
+ ]
+ },
+ "execution_count": 167,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "dom(h)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "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": 14,
+ "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,
+ "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=\"206pt\" height=\"314pt\"\n",
+ " viewBox=\"0.00 0.00 206.00 314.00\" 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(4 310)\">\n",
+ "<title>state</title>\n",
+ "<polygon fill=\"white\" stroke=\"white\" points=\"-4,5 -4,-310 203,-310 203,5 -4,5\"/>\n",
+ "<!-- 10 -->\n",
+ "<g id=\"node1\" class=\"node\"><title>10</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-306 0,-306 0,-270 54,-270 54,-306\"/>\n",
+ "<text text-anchor=\"middle\" x=\"27\" y=\"-283.8\" font-family=\"Times,serif\" font-size=\"14.00\">10</text>\n",
+ "</g>\n",
+ "<!-- 5 -->\n",
+ "<g id=\"node3\" class=\"node\"><title>5</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\">5</text>\n",
+ "</g>\n",
+ "<!-- 10->5 -->\n",
+ "<g id=\"edge2\" class=\"edge\"><title>10->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-269.614C27,-257.24 27,-240.369 27,-226.22\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5001,-226.05 27,-216.05 23.5001,-226.05 30.5001,-226.05\"/>\n",
+ "<text text-anchor=\"middle\" x=\"30.5\" y=\"-238.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "</g>\n",
+ "<!-- 8 -->\n",
+ "<g id=\"node4\" class=\"node\"><title>8</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"126,-306 72,-306 72,-270 126,-270 126,-306\"/>\n",
+ "<text text-anchor=\"middle\" x=\"99\" y=\"-283.8\" font-family=\"Times,serif\" font-size=\"14.00\">8</text>\n",
+ "</g>\n",
+ "<!-- 4 -->\n",
+ "<g id=\"node6\" class=\"node\"><title>4</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"126,-216 72,-216 72,-180 126,-180 126,-216\"/>\n",
+ "<text text-anchor=\"middle\" x=\"99\" y=\"-193.8\" font-family=\"Times,serif\" font-size=\"14.00\">4</text>\n",
+ "</g>\n",
+ "<!-- 8->4 -->\n",
+ "<g id=\"edge4\" class=\"edge\"><title>8->4</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-269.614C99,-257.24 99,-240.369 99,-226.22\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-226.05 99,-216.05 95.5001,-226.05 102.5,-226.05\"/>\n",
+ "<text text-anchor=\"middle\" x=\"102.5\" y=\"-238.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "</g>\n",
+ "<!-- 2 -->\n",
+ "<g id=\"node11\" class=\"node\"><title>2</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"126,-126 72,-126 72,-90 126,-90 126,-126\"/>\n",
+ "<text text-anchor=\"middle\" x=\"99\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
+ "</g>\n",
+ "<!-- 4->2 -->\n",
+ "<g id=\"edge8\" class=\"edge\"><title>4->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-179.614C99,-167.24 99,-150.369 99,-136.22\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-136.05 99,-126.05 95.5001,-136.05 102.5,-136.05\"/>\n",
+ "<text text-anchor=\"middle\" x=\"102.5\" y=\"-148.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "</g>\n",
+ "<!-- 6 -->\n",
+ "<g id=\"node7\" class=\"node\"><title>6</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"198,-306 144,-306 144,-270 198,-270 198,-306\"/>\n",
+ "<text text-anchor=\"middle\" x=\"171\" y=\"-283.8\" font-family=\"Times,serif\" font-size=\"14.00\">6</text>\n",
+ "</g>\n",
+ "<!-- 3 -->\n",
+ "<g id=\"node9\" class=\"node\"><title>3</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"198,-216 144,-216 144,-180 198,-180 198,-216\"/>\n",
+ "<text text-anchor=\"middle\" x=\"171\" y=\"-193.8\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
+ "</g>\n",
+ "<!-- 6->3 -->\n",
+ "<g id=\"edge6\" class=\"edge\"><title>6->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M171,-269.614C171,-257.24 171,-240.369 171,-226.22\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"174.5,-226.05 171,-216.05 167.5,-226.05 174.5,-226.05\"/>\n",
+ "<text text-anchor=\"middle\" x=\"174.5\" y=\"-238.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "</g>\n",
+ "<!-- 1 -->\n",
+ "<g id=\"node13\" class=\"node\"><title>1</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"126,-36 72,-36 72,-0 126,-0 126,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"99\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
+ "</g>\n",
+ "<!-- 2->1 -->\n",
+ "<g id=\"edge10\" class=\"edge\"><title>2->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-89.614C99,-77.2403 99,-60.3686 99,-46.2198\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-46.0504 99,-36.0504 95.5001,-46.0504 102.5,-46.0504\"/>\n",
+ "<text text-anchor=\"middle\" x=\"102.5\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "</g>\n",
+ "</g>\n",
+ "</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),(4,2),(6,3),(8,4),(10,5)}(\"h\",h)]>"
+ ]
+ },
+ "execution_count": 15,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":dot expr_as_graph (\"h\",h)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 9,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{(1\\mapsto\\text{\"nein\"}),(2\\mapsto\\text{\"nein\"}),(3\\mapsto\\text{\"ja\"}),(4\\mapsto\\text{\"nein\"}),(5\\mapsto\\text{\"nein\"}),(6\\mapsto\\text{\"ja\"}),(7\\mapsto\\text{\"nein\"})\\}$"
+ ],
+ "text/plain": [
+ "{(1↦\"nein\"),(2↦\"nein\"),(3↦\"ja\"),(4↦\"nein\"),(5↦\"nein\"),(6↦\"ja\"),(7↦\"nein\")}"
+ ]
+ },
+ "execution_count": 9,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":let d {x,y| x:1..7 ∧ (x mod 3 =0 => y = \"ja\") ∧ (x mod 3 >0 => y= \"nein\")}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{1,2,3,4,5,6,7\\}$"
+ ],
+ "text/plain": [
+ "{1,2,3,4,5,6,7}"
+ ]
+ },
+ "execution_count": 10,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "dom(d)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{\\text{\"ja\"},\\text{\"nein\"}\\}$"
+ ],
+ "text/plain": [
+ "{\"ja\",\"nein\"}"
+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "ran(d)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "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=\"494pt\" height=\"134pt\"\n",
+ " viewBox=\"0.00 0.00 494.00 134.00\" 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(4 130)\">\n",
+ "<title>state</title>\n",
+ "<polygon fill=\"white\" stroke=\"white\" points=\"-4,5 -4,-130 491,-130 491,5 -4,5\"/>\n",
+ "<!-- 7 -->\n",
+ "<g id=\"node1\" class=\"node\"><title>7</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-126 0,-126 0,-90 54,-90 54,-126\"/>\n",
+ "<text text-anchor=\"middle\" x=\"27\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">7</text>\n",
+ "</g>\n",
+ "<!-- "nein" -->\n",
+ "<g id=\"node3\" class=\"node\"><title>"nein"</title>\n",
+ "<polygon fill=\"#fff68f\" stroke=\"#fff68f\" points=\"198,-36 144,-36 144,-0 198,-0 198,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"171\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">"nein"</text>\n",
+ "</g>\n",
+ "<!-- 7->"nein" -->\n",
+ "<g id=\"edge2\" class=\"edge\"><title>7->"nein"</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.0853,-90.4479C76.8692,-76.5243 109.644,-56.4953 134.698,-41.1844\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"136.839,-43.9781 143.547,-35.7771 133.189,-38.0052 136.839,-43.9781\"/>\n",
+ "<text text-anchor=\"middle\" x=\"115.5\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
+ "</g>\n",
+ "<!-- 6 -->\n",
+ "<g id=\"node4\" class=\"node\"><title>6</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"414,-126 360,-126 360,-90 414,-90 414,-126\"/>\n",
+ "<text text-anchor=\"middle\" x=\"387\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">6</text>\n",
+ "</g>\n",
+ "<!-- "ja" -->\n",
+ "<g id=\"node6\" class=\"node\"><title>"ja"</title>\n",
+ "<polygon fill=\"#fff68f\" stroke=\"#fff68f\" points=\"450,-36 396,-36 396,-0 450,-0 450,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"423\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">"ja"</text>\n",
+ "</g>\n",
+ "<!-- 6->"ja" -->\n",
+ "<g id=\"edge4\" class=\"edge\"><title>6->"ja"</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M394.112,-89.614C399.274,-76.9977 406.348,-59.7052 412.204,-45.3907\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"415.478,-46.6311 416.025,-36.0504 408.999,-43.9807 415.478,-46.6311\"/>\n",
+ "<text text-anchor=\"middle\" x=\"411.5\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
+ "</g>\n",
+ "<!-- 5 -->\n",
+ "<g id=\"node7\" class=\"node\"><title>5</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"126,-126 72,-126 72,-90 126,-90 126,-126\"/>\n",
+ "<text text-anchor=\"middle\" x=\"99\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">5</text>\n",
+ "</g>\n",
+ "<!-- 5->"nein" -->\n",
+ "<g id=\"edge6\" class=\"edge\"><title>5->"nein"</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M113.225,-89.614C123.845,-76.6338 138.515,-58.7037 150.415,-44.1595\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"153.426,-46.0064 157.05,-36.0504 148.008,-41.5737 153.426,-46.0064\"/>\n",
+ "<text text-anchor=\"middle\" x=\"145.5\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
+ "</g>\n",
+ "<!-- 4 -->\n",
+ "<g id=\"node9\" class=\"node\"><title>4</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"198,-126 144,-126 144,-90 198,-90 198,-126\"/>\n",
+ "<text text-anchor=\"middle\" x=\"171\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">4</text>\n",
+ "</g>\n",
+ "<!-- 4->"nein" -->\n",
+ "<g id=\"edge8\" class=\"edge\"><title>4->"nein"</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M171,-89.614C171,-77.2403 171,-60.3686 171,-46.2198\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"174.5,-46.0504 171,-36.0504 167.5,-46.0504 174.5,-46.0504\"/>\n",
+ "<text text-anchor=\"middle\" x=\"174.5\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
+ "</g>\n",
+ "<!-- 3 -->\n",
+ "<g id=\"node11\" class=\"node\"><title>3</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"486,-126 432,-126 432,-90 486,-90 486,-126\"/>\n",
+ "<text text-anchor=\"middle\" x=\"459\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
+ "</g>\n",
+ "<!-- 3->"ja" -->\n",
+ "<g id=\"edge10\" class=\"edge\"><title>3->"ja"</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M451.888,-89.614C446.726,-76.9977 439.652,-59.7052 433.796,-45.3907\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"437.001,-43.9807 429.975,-36.0504 430.522,-46.6311 437.001,-43.9807\"/>\n",
+ "<text text-anchor=\"middle\" x=\"447.5\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
+ "</g>\n",
+ "<!-- 2 -->\n",
+ "<g id=\"node13\" class=\"node\"><title>2</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"270,-126 216,-126 216,-90 270,-90 270,-126\"/>\n",
+ "<text text-anchor=\"middle\" x=\"243\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
+ "</g>\n",
+ "<!-- 2->"nein" -->\n",
+ "<g id=\"edge12\" class=\"edge\"><title>2->"nein"</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M228.775,-89.614C218.155,-76.6338 203.485,-58.7037 191.585,-44.1595\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"193.992,-41.5737 184.95,-36.0504 188.574,-46.0064 193.992,-41.5737\"/>\n",
+ "<text text-anchor=\"middle\" x=\"217.5\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
+ "</g>\n",
+ "<!-- 1 -->\n",
+ "<g id=\"node15\" class=\"node\"><title>1</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"342,-126 288,-126 288,-90 342,-90 342,-126\"/>\n",
+ "<text text-anchor=\"middle\" x=\"315\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
+ "</g>\n",
+ "<!-- 1->"nein" -->\n",
+ "<g id=\"edge14\" class=\"edge\"><title>1->"nein"</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M287.915,-90.4479C265.131,-76.5243 232.356,-56.4953 207.302,-41.1844\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"208.811,-38.0052 198.453,-35.7771 205.161,-43.9781 208.811,-38.0052\"/>\n",
+ "<text text-anchor=\"middle\" x=\"259.5\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
+ "</g>\n",
+ "</g>\n",
+ "</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)]>"
+ ]
+ },
+ "execution_count": 12,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":dot expr_as_graph (\"d\",d)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Relationales Abbild und Umkehrrelation\n",
+ " \n",
+ "Abbild:\n",
+ " * $r[A] = \\{b \\mid \\exists a.((a,b)\\in r \\wedge a\\in A)\\}$\n",
+ "\n",
+ "Umkehrrelation:\n",
+ " * $r^{-1} = \\{(b,a) \\mid (a,b)\\in r \\}$\\\\\n",
+ "\n",
+ "Im Notebook muss für die Umkehrrelation leider Tilde (~) verwendet werden: "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{4,5\\}$"
+ ],
+ "text/plain": [
+ "{4,5}"
+ ]
+ },
+ "execution_count": 16,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "h[8..10]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 17,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{\\text{\"ja\"}\\}$"
+ ],
+ "text/plain": [
+ "{\"ja\"}"
+ ]
+ },
+ "execution_count": 17,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "d[{3,6}]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 19,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$"
+ ],
+ "text/plain": [
+ "∅"
+ ]
+ },
+ "execution_count": 19,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "d[{0}]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 26,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{(\\text{\"ja\"}\\mapsto 3),(\\text{\"ja\"}\\mapsto 6),(\\text{\"nein\"}\\mapsto 1),(\\text{\"nein\"}\\mapsto 2),(\\text{\"nein\"}\\mapsto 4),(\\text{\"nein\"}\\mapsto 5),(\\text{\"nein\"}\\mapsto 7)\\}$"
+ ],
+ "text/plain": [
+ "{(\"ja\"↦3),(\"ja\"↦6),(\"nein\"↦1),(\"nein\"↦2),(\"nein\"↦4),(\"nein\"↦5),(\"nein\"↦7)}"
+ ]
+ },
+ "execution_count": 26,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "d~"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 38,
+ "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=\"494pt\" height=\"134pt\"\n",
+ " viewBox=\"0.00 0.00 494.00 134.00\" 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(4 130)\">\n",
+ "<title>state</title>\n",
+ "<polygon fill=\"white\" stroke=\"white\" points=\"-4,5 -4,-130 491,-130 491,5 -4,5\"/>\n",
+ "<!-- "nein" -->\n",
+ "<g id=\"node1\" class=\"node\"><title>"nein"</title>\n",
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+ "</g>\n",
+ "<!-- 7 -->\n",
+ "<g id=\"node3\" class=\"node\"><title>7</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-36 0,-36 0,-0 54,-0 54,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"27\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">7</text>\n",
+ "</g>\n",
+ "<!-- "nein"->7 -->\n",
+ "<g id=\"edge2\" class=\"edge\"><title>"nein"->7</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M143.719,-91.5608C133.251,-85.5518 121.229,-78.5478 110.427,-72 94.5084,-62.3502 77.0995,-51.3612 62.3222,-41.901\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"63.8037,-38.6929 53.4982,-36.2339 60.021,-44.5828 63.8037,-38.6929\"/>\n",
+ "<text text-anchor=\"middle\" x=\"118.286\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">d~</text>\n",
+ "</g>\n",
+ "<!-- 5 -->\n",
+ "<g id=\"node5\" class=\"node\"><title>5</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"126,-36 72,-36 72,-0 126,-0 126,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"99\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">5</text>\n",
+ "</g>\n",
+ "<!-- "nein"->5 -->\n",
+ "<g id=\"edge4\" class=\"edge\"><title>"nein"->5</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M156.775,-89.614C146.155,-76.6338 131.485,-58.7037 119.585,-44.1595\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"121.992,-41.5737 112.95,-36.0504 116.574,-46.0064 121.992,-41.5737\"/>\n",
+ "<text text-anchor=\"middle\" x=\"149.286\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">d~</text>\n",
+ "</g>\n",
+ "<!-- 4 -->\n",
+ "<g id=\"node7\" class=\"node\"><title>4</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"198,-36 144,-36 144,-0 198,-0 198,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"171\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">4</text>\n",
+ "</g>\n",
+ "<!-- "nein"->4 -->\n",
+ "<g id=\"edge6\" class=\"edge\"><title>"nein"->4</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M171,-89.614C171,-77.2403 171,-60.3686 171,-46.2198\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"174.5,-46.0504 171,-36.0504 167.5,-46.0504 174.5,-46.0504\"/>\n",
+ "<text text-anchor=\"middle\" x=\"178.286\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">d~</text>\n",
+ "</g>\n",
+ "<!-- 2 -->\n",
+ "<g id=\"node9\" class=\"node\"><title>2</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"270,-36 216,-36 216,-0 270,-0 270,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"243\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
+ "</g>\n",
+ "<!-- "nein"->2 -->\n",
+ "<g id=\"edge8\" class=\"edge\"><title>"nein"->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M185.225,-89.614C195.845,-76.6338 210.515,-58.7037 222.415,-44.1595\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"225.426,-46.0064 229.05,-36.0504 220.008,-41.5737 225.426,-46.0064\"/>\n",
+ "<text text-anchor=\"middle\" x=\"221.286\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">d~</text>\n",
+ "</g>\n",
+ "<!-- 1 -->\n",
+ "<g id=\"node11\" class=\"node\"><title>1</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"342,-36 288,-36 288,-0 342,-0 342,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"315\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
+ "</g>\n",
+ "<!-- "nein"->1 -->\n",
+ "<g id=\"edge10\" class=\"edge\"><title>"nein"->1</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M198.187,-91.7663C208.802,-85.7232 221.034,-78.6418 232,-72 247.985,-62.3177 265.453,-51.2405 280.222,-41.7252\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"282.534,-44.3978 289.031,-36.0288 278.733,-38.5196 282.534,-44.3978\"/>\n",
+ "<text text-anchor=\"middle\" x=\"267.286\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">d~</text>\n",
+ "</g>\n",
+ "<!-- "ja" -->\n",
+ "<g id=\"node12\" class=\"node\"><title>"ja"</title>\n",
+ "<polygon fill=\"#fff68f\" stroke=\"#fff68f\" points=\"450,-126 396,-126 396,-90 450,-90 450,-126\"/>\n",
+ "<text text-anchor=\"middle\" x=\"423\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">"ja"</text>\n",
+ "</g>\n",
+ "<!-- 6 -->\n",
+ "<g id=\"node14\" class=\"node\"><title>6</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"414,-36 360,-36 360,-0 414,-0 414,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"387\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">6</text>\n",
+ "</g>\n",
+ "<!-- "ja"->6 -->\n",
+ "<g id=\"edge12\" class=\"edge\"><title>"ja"->6</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M415.888,-89.614C410.726,-76.9977 403.652,-59.7052 397.796,-45.3907\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"401.001,-43.9807 393.975,-36.0504 394.522,-46.6311 401.001,-43.9807\"/>\n",
+ "<text text-anchor=\"middle\" x=\"416.286\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">d~</text>\n",
+ "</g>\n",
+ "<!-- 3 -->\n",
+ "<g id=\"node16\" class=\"node\"><title>3</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"486,-36 432,-36 432,-0 486,-0 486,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"459\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
+ "</g>\n",
+ "<!-- "ja"->3 -->\n",
+ "<g id=\"edge14\" class=\"edge\"><title>"ja"->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M430.112,-89.614C435.274,-76.9977 442.348,-59.7052 448.204,-45.3907\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"451.478,-46.6311 452.025,-36.0504 444.999,-43.9807 451.478,-46.6311\"/>\n",
+ "<text text-anchor=\"middle\" x=\"452.286\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">d~</text>\n",
+ "</g>\n",
+ "</g>\n",
+ "</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),(4,2),(6,3),(8,4),(10,5)}(\"d~\",d~)]>"
+ ]
+ },
+ "execution_count": 38,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":dot expr_as_graph (\"d~\",d~)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Man kann natürlich auch beide Operatore verknüpfen, zum Beispiel um herauszufinden welche Zahlen durch 3 teilbar sind:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 40,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{3,6\\}$"
+ ],
+ "text/plain": [
+ "{3,6}"
+ ]
+ },
+ "execution_count": 40,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "d~[{\"ja\"}]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 54,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ ":unlet d"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 32,
+ "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",
+ "Wir können Relationen $r_1$ und $r_2$ mit dem Operator ``;'' verknüpfen:\n",
+ "* $(r_1 ; r_2)$ = $\\{(a,c) \\mid \\exists b. ( (a,b)\\in r_1 \\wedge (b,c)\\in r_2)\\}$.\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 42,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{(1\\mapsto 4),(1\\mapsto 8)\\}$"
+ ],
+ "text/plain": [
+ "{(1↦4),(1↦8)}"
+ ]
+ },
+ "execution_count": 42,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "({1↦2, 2↦3} ; {2↦4, 2↦8})"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "Man kann eine Relation auch mit sich selber verknüpfen wenn Werte- und Definitionsbereich kompatibel sind.\n",
+ "Zum Beispiel $h^2 = (h;h)$:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 49,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{(4\\mapsto 1),(8\\mapsto 2)\\}$"
+ ],
+ "text/plain": [
+ "{(4↦1),(8↦2)}"
+ ]
+ },
+ "execution_count": 49,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "(h ; h)"
+ ]
+ },
+ {
+ "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",
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+ "<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",
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+ "<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",
+ "<text text-anchor=\"middle\" x=\"631\" y=\"-124.2\" font-family=\"Times,serif\" font-size=\"14.00\">8</text>\n",
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+ "metadata": {},
+ "source": [
+ "# Transitive und Reflexive Hülle\n",
+ " \n",
+ "Gegeben eine Relation $r$ von $A$ nach $A$\n",
+ "* $r^0 = \\{(a,a) \\mid a\\in A\\}$\n",
+ "* $r^1 = r$\n",
+ "* $r^k = (r^{k-1} ; r) = (r ; r^{k-1})$\n",
+ "* $r^*$ = $\\bigcup_{i\\geq 0} r^i$\n",
+ "\n",
+ "Die transitive Hülle von $r$ ist:\n",
+ "* $r^+$ = $\\bigcup_{i\\geq 1} r^i$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 51,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{1,2,3\\}$"
+ ],
+ "text/plain": [
+ "{1,2,3}"
+ ]
+ },
+ "execution_count": 51,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":let A {1,2,3}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 52,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{(1\\mapsto 2),(2\\mapsto 3)\\}$"
+ ],
+ "text/plain": [
+ "{(1↦2),(2↦3)}"
+ ]
+ },
+ "execution_count": 52,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":let r {(1,2), (2,3)}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 62,
+ "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=\"62pt\" height=\"224pt\"\n",
+ " viewBox=\"0.00 0.00 62.00 224.00\" 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 59,-220 59,5 -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",
+ "<text text-anchor=\"middle\" x=\"27\" y=\"-103.8\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
+ "</g>\n",
+ "<!-- 3 -->\n",
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+ "<text text-anchor=\"middle\" x=\"27\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
+ "</g>\n",
+ "<!-- 2->3 -->\n",
+ "<g id=\"edge2\" class=\"edge\"><title>2->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-89.614C27,-77.2403 27,-60.3686 27,-46.2198\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5001,-46.0504 27,-36.0504 23.5001,-46.0504 30.5001,-46.0504\"/>\n",
+ "<text text-anchor=\"middle\" x=\"29.3303\" y=\"-58.8\" font-family=\"Times,serif\" font-size=\"14.00\">r</text>\n",
+ "</g>\n",
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+ "<g id=\"node4\" class=\"node\"><title>1</title>\n",
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+ "</g>\n",
+ "<!-- 1->2 -->\n",
+ "<g id=\"edge4\" class=\"edge\"><title>1->2</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-179.614C27,-167.24 27,-150.369 27,-136.22\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5001,-136.05 27,-126.05 23.5001,-136.05 30.5001,-136.05\"/>\n",
+ "<text text-anchor=\"middle\" x=\"29.3303\" y=\"-148.8\" 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\",r)]>"
+ ]
+ },
+ "execution_count": 62,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":dot expr_as_graph (\"r\",r)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 56,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{(1\\mapsto 3)\\}$"
+ ],
+ "text/plain": [
+ "{(1↦3)}"
+ ]
+ },
+ "execution_count": 56,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "(r;r)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "In B kann man $r^0$ auch als die Identitätsrelation über A beschreiben:"
+ ]
+ },
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 67,
"metadata": {},
- "outputs": [],
- "source": []
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{(1\\mapsto 1),(2\\mapsto 2),(3\\mapsto 3)\\}$"
+ ],
+ "text/plain": [
+ "{(1↦1),(2↦2),(3↦3)}"
+ ]
+ },
+ "execution_count": 67,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "id(A)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Die transitive und reflexive Hülle ist hier:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 60,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{(1\\mapsto 1),(1\\mapsto 2),(1\\mapsto 3),(2\\mapsto 2),(2\\mapsto 3),(3\\mapsto 3)\\}$"
+ ],
+ "text/plain": [
+ "{(1↦1),(1↦2),(1↦3),(2↦2),(2↦3),(3↦3)}"
+ ]
+ },
+ "execution_count": 60,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "id(A) \\/ r \\/ (r;r) \\/ (r;r;r) \\/ (r;r;r;r)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 61,
+ "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",
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+ " -->\n",
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+ "<svg width=\"127pt\" height=\"224pt\"\n",
+ " viewBox=\"0.00 0.00 126.66 224.00\" 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,-220 123.661,-220 123.661,5 -4,5\"/>\n",
+ "<!-- 3 -->\n",
+ "<g id=\"node1\" class=\"node\"><title>3</title>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-36 0,-36 0,-0 54,-0 54,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"27\" y=\"-13.8\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
+ "</g>\n",
+ "<!-- 3->3 -->\n",
+ "<g id=\"edge2\" class=\"edge\"><title>3->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.2408,-26.2419C64.0239,-26.4192 72,-23.6719 72,-18 72,-14.5437 69.0382,-12.1734 64.5105,-10.8891\"/>\n",
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+ "</g>\n",
+ "<!-- 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",
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+ "</g>\n",
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+ "<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",
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+ "</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",
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+ "</g>\n",
+ "<!-- 1->2 -->\n",
+ "<g id=\"edge10\" class=\"edge\"><title>1->2</title>\n",
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+ "</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",
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+ "</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": [
+ "Die transitive Hülle wird in B als ```closure1``` geschrieben:"
+ ]
+ },
+ {
+ "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",
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+ "</g>\n",
+ "<!-- 2->3 -->\n",
+ "<g id=\"edge2\" class=\"edge\"><title>2->3</title>\n",
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+ "</g>\n",
+ "<!-- 1->2 -->\n",
+ "<g id=\"edge6\" class=\"edge\"><title>1->2</title>\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>\n",
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+ "<g id=\"edge4\" class=\"edge\"><title>1->3</title>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M67,-179.933C67,-148.966 67,-83.2989 67,-46.1292\"/>\n",
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+ "</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+\",closure1(r))]>"
+ ]
+ },
+ "execution_count": 65,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":dot expr_as_graph (\"r+\",closure1(r))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Man kann die transitive Hülle natürlich mit den anderen Operatoren verknüpfen, zum Beispiel um auszurechnen von welchen Knoten aus man den Knoten 3 erreichen kann:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 69,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{1,2\\}$"
+ ],
+ "text/plain": [
+ "{1,2}"
+ ]
+ },
+ "execution_count": 69,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "closure1(r)~[{3}]"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Anmerkung: die reflexive und transitive Hülle in B wird als ```closure``` geschrieben, ist bei Relationen über Zahlen immer unendlich."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 72,
+ "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"
+ }
+ ],
+ "source": [
+ "closure(r)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 71,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$\\{1,2,3\\}$"
+ ],
+ "text/plain": [
+ "{1,2,3}"
+ ]
+ },
+ "execution_count": 71,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "closure(r)~[{3}]"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Funktionen\n",
+ " \n",
+ "Was unterscheidet Funktionen von Relationen?\n",
+ "Wie kann man Funktionen in Mengentheorie und Logik darstellen?"
+ ]
}
],
"metadata": {