Mathematik1_2019-11-26.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1. Schnittmengen"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = {1, 2, 3, 4, 5}\n",
    "B = {4, 5, 6, 7, 8}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1, 2, 3, 4, 5, 6, 7, 8}"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A & B\n",
    "A.copy().union(B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{4, 5}"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A | B \n",
    "{x for x in A if x in B}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1, 2, 3}"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A\\B\n",
    "{x for x in A if x not in B}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2. Menge Natürlicher zahlen zwischen 11 und 50,000 (exklusiv) die gerade sind"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{12,\n",
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       " 1732,\n",
       " 1734,\n",
       " 1736,\n",
       " 1738,\n",
       " 1740,\n",
       " 1742,\n",
       " 1744,\n",
       " 1746,\n",
       " 1748,\n",
       " 1750,\n",
       " 1752,\n",
       " 1754,\n",
       " 1756,\n",
       " 1758,\n",
       " 1760,\n",
       " 1762,\n",
       " 1764,\n",
       " 1766,\n",
       " 1768,\n",
       " 1770,\n",
       " 1772,\n",
       " 1774,\n",
       " 1776,\n",
       " 1778,\n",
       " 1780,\n",
       " 1782,\n",
       " 1784,\n",
       " 1786,\n",
       " 1788,\n",
       " 1790,\n",
       " 1792,\n",
       " 1794,\n",
       " 1796,\n",
       " 1798,\n",
       " 1800,\n",
       " 1802,\n",
       " 1804,\n",
       " 1806,\n",
       " 1808,\n",
       " 1810,\n",
       " 1812,\n",
       " 1814,\n",
       " 1816,\n",
       " 1818,\n",
       " 1820,\n",
       " 1822,\n",
       " 1824,\n",
       " 1826,\n",
       " 1828,\n",
       " 1830,\n",
       " 1832,\n",
       " 1834,\n",
       " 1836,\n",
       " 1838,\n",
       " 1840,\n",
       " 1842,\n",
       " 1844,\n",
       " 1846,\n",
       " 1848,\n",
       " 1850,\n",
       " 1852,\n",
       " 1854,\n",
       " 1856,\n",
       " 1858,\n",
       " 1860,\n",
       " 1862,\n",
       " 1864,\n",
       " 1866,\n",
       " 1868,\n",
       " 1870,\n",
       " 1872,\n",
       " 1874,\n",
       " 1876,\n",
       " 1878,\n",
       " 1880,\n",
       " 1882,\n",
       " 1884,\n",
       " 1886,\n",
       " 1888,\n",
       " 1890,\n",
       " 1892,\n",
       " 1894,\n",
       " 1896,\n",
       " 1898,\n",
       " 1900,\n",
       " 1902,\n",
       " 1904,\n",
       " 1906,\n",
       " 1908,\n",
       " 1910,\n",
       " 1912,\n",
       " 1914,\n",
       " 1916,\n",
       " 1918,\n",
       " 1920,\n",
       " 1922,\n",
       " 1924,\n",
       " 1926,\n",
       " 1928,\n",
       " 1930,\n",
       " 1932,\n",
       " 1934,\n",
       " 1936,\n",
       " 1938,\n",
       " 1940,\n",
       " 1942,\n",
       " 1944,\n",
       " 1946,\n",
       " 1948,\n",
       " 1950,\n",
       " 1952,\n",
       " 1954,\n",
       " 1956,\n",
       " 1958,\n",
       " 1960,\n",
       " 1962,\n",
       " 1964,\n",
       " 1966,\n",
       " 1968,\n",
       " 1970,\n",
       " 1972,\n",
       " 1974,\n",
       " 1976,\n",
       " 1978,\n",
       " 1980,\n",
       " 1982,\n",
       " 1984,\n",
       " 1986,\n",
       " 1988,\n",
       " 1990,\n",
       " 1992,\n",
       " 1994,\n",
       " 1996,\n",
       " 1998,\n",
       " 2000,\n",
       " 2002,\n",
       " 2004,\n",
       " 2006,\n",
       " 2008,\n",
       " 2010,\n",
       " ...}"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{n for n in range(12, 50000) if n % 2 == 0}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3. Tupel als Elemente einer Menge"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = (1,2,3)\n",
    "B = (2,3,4)\n",
    "C = {A,B}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(True, True, False, True)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 in A, A in C, 1 in C, 1 in A in C"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4. Kartesisches Produkt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(1, 'a'), (1, 'b'), (2, 'a'), (2, 'b'), (3, 'a'), (3, 'b')}"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A x B\n",
    "A = {1, 2, 3}\n",
    "B = {'a', 'b'}\n",
    "{(a, b) for a in A for b in B}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "({(1, 'a'), (1, 'b'), (2, 'a'), (2, 'b'), (3, 'a'), (3, 'b')},\n",
       " {('a', 1), ('a', 2), ('a', 3), ('b', 1), ('b', 2), ('b', 3)},\n",
       " False)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A x B != B x A\n",
    "AxB = {(a, b) for a in A for b in B}\n",
    "BxA = {(b, a) for a in A for b in B}\n",
    "\n",
    "AxB, BxA, AxB == BxA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{((1, 1), (1, 1)),\n",
       " ((1, 1), (2, 2)),\n",
       " ((1, 2), (1, 1)),\n",
       " ((1, 2), (2, 2)),\n",
       " ((2, 1), (1, 1)),\n",
       " ((2, 1), (2, 2)),\n",
       " ((2, 2), (1, 1)),\n",
       " ((2, 2), (2, 2))}"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A^3\n",
    "A = {1, 2}\n",
    "{(a2, a3) for a2 in {(a, a1) for a in A for a1 in A} for a3 in {(a, a) for a in A for a1 in A}}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1000,\n",
       " 1001,\n",
       " 1004,\n",
       " 1005,\n",
       " 1006,\n",
       " 1007,\n",
       " 1008,\n",
       " 1009,\n",
       " 1020,\n",
       " 1021,\n",
       " 1024,\n",
       " 1025,\n",
       " 1026,\n",
       " 1027,\n",
       " 1028,\n",
       " 1029,\n",
       " 1030,\n",
       " 1031,\n",
       " 1034,\n",
       " 1035,\n",
       " 1036,\n",
       " 1037,\n",
       " 1038,\n",
       " 1039,\n",
       " 1040,\n",
       " 1041,\n",
       " 1044,\n",
       " 1045,\n",
       " 1046,\n",
       " 1047,\n",
       " 1048,\n",
       " 1049,\n",
       " 1060,\n",
       " 1061,\n",
       " 1064,\n",
       " 1065,\n",
       " 1066,\n",
       " 1067,\n",
       " 1068,\n",
       " 1069,\n",
       " 1070,\n",
       " 1071,\n",
       " 1074,\n",
       " 1075,\n",
       " 1076,\n",
       " 1077,\n",
       " 1078,\n",
       " 1079,\n",
       " 1080,\n",
       " 1081,\n",
       " 1084,\n",
       " 1085,\n",
       " 1086,\n",
       " 1087,\n",
       " 1088,\n",
       " 1089,\n",
       " 1090,\n",
       " 1091,\n",
       " 1094,\n",
       " 1095,\n",
       " 1096,\n",
       " 1097,\n",
       " 1098,\n",
       " 1099,\n",
       " 1100,\n",
       " 1101,\n",
       " 1104,\n",
       " 1105,\n",
       " 1106,\n",
       " 1107,\n",
       " 1108,\n",
       " 1109,\n",
       " 1120,\n",
       " 1121,\n",
       " 1124,\n",
       " 1125,\n",
       " 1126,\n",
       " 1127,\n",
       " 1128,\n",
       " 1129,\n",
       " 1130,\n",
       " 1131,\n",
       " 1134,\n",
       " 1135,\n",
       " 1136,\n",
       " 1137,\n",
       " 1138,\n",
       " 1139,\n",
       " 1140,\n",
       " 1141,\n",
       " 1144,\n",
       " 1145,\n",
       " 1146,\n",
       " 1147,\n",
       " 1148,\n",
       " 1149,\n",
       " 1160,\n",
       " 1161,\n",
       " 1164,\n",
       " 1165,\n",
       " 1166,\n",
       " 1167,\n",
       " 1168,\n",
       " 1169,\n",
       " 1170,\n",
       " 1171,\n",
       " 1174,\n",
       " 1175,\n",
       " 1176,\n",
       " 1177,\n",
       " 1178,\n",
       " 1179,\n",
       " 1180,\n",
       " 1181,\n",
       " 1184,\n",
       " 1185,\n",
       " 1186,\n",
       " 1187,\n",
       " 1188,\n",
       " 1189,\n",
       " 1190,\n",
       " 1191,\n",
       " 1194,\n",
       " 1195,\n",
       " 1196,\n",
       " 1197,\n",
       " 1198,\n",
       " 1199,\n",
       " 1200,\n",
       " 1201,\n",
       " 1204,\n",
       " 1205,\n",
       " 1206,\n",
       " 1207,\n",
       " 1208,\n",
       " 1209,\n",
       " 1220,\n",
       " 1221,\n",
       " 1224,\n",
       " 1225,\n",
       " 1226,\n",
       " 1227,\n",
       " 1228,\n",
       " 1229,\n",
       " 1230,\n",
       " 1231,\n",
       " 1234,\n",
       " 1235,\n",
       " 1236,\n",
       " 1237,\n",
       " 1238,\n",
       " 1239,\n",
       " 1240,\n",
       " 1241,\n",
       " 1244,\n",
       " 1245,\n",
       " 1246,\n",
       " 1247,\n",
       " 1248,\n",
       " 1249,\n",
       " 1260,\n",
       " 1261,\n",
       " 1264,\n",
       " 1265,\n",
       " 1266,\n",
       " 1267,\n",
       " 1268,\n",
       " 1269,\n",
       " 1270,\n",
       " 1271,\n",
       " 1274,\n",
       " 1275,\n",
       " 1276,\n",
       " 1277,\n",
       " 1278,\n",
       " 1279,\n",
       " 1280,\n",
       " 1281,\n",
       " 1284,\n",
       " 1285,\n",
       " 1286,\n",
       " 1287,\n",
       " 1288,\n",
       " 1289,\n",
       " 1290,\n",
       " 1291,\n",
       " 1294,\n",
       " 1295,\n",
       " 1296,\n",
       " 1297,\n",
       " 1298,\n",
       " 1299,\n",
       " 1300,\n",
       " 1301,\n",
       " 1304,\n",
       " 1305,\n",
       " 1306,\n",
       " 1307,\n",
       " 1308,\n",
       " 1309,\n",
       " 1320,\n",
       " 1321,\n",
       " 1324,\n",
       " 1325,\n",
       " 1326,\n",
       " 1327,\n",
       " 1328,\n",
       " 1329,\n",
       " 1330,\n",
       " 1331,\n",
       " 1334,\n",
       " 1335,\n",
       " 1336,\n",
       " 1337,\n",
       " 1338,\n",
       " 1339,\n",
       " 1340,\n",
       " 1341,\n",
       " 1344,\n",
       " 1345,\n",
       " 1346,\n",
       " 1347,\n",
       " 1348,\n",
       " 1349,\n",
       " 1360,\n",
       " 1361,\n",
       " 1364,\n",
       " 1365,\n",
       " 1366,\n",
       " 1367,\n",
       " 1368,\n",
       " 1369,\n",
       " 1370,\n",
       " 1371,\n",
       " 1374,\n",
       " 1375,\n",
       " 1376,\n",
       " 1377,\n",
       " 1378,\n",
       " 1379,\n",
       " 1380,\n",
       " 1381,\n",
       " 1384,\n",
       " 1385,\n",
       " 1386,\n",
       " 1387,\n",
       " 1388,\n",
       " 1389,\n",
       " 1390,\n",
       " 1391,\n",
       " 1394,\n",
       " 1395,\n",
       " 1396,\n",
       " 1397,\n",
       " 1398,\n",
       " 1399,\n",
       " 1500,\n",
       " 1501,\n",
       " 1504,\n",
       " 1505,\n",
       " 1506,\n",
       " 1507,\n",
       " 1508,\n",
       " 1509,\n",
       " 1520,\n",
       " 1521,\n",
       " 1524,\n",
       " 1525,\n",
       " 1526,\n",
       " 1527,\n",
       " 1528,\n",
       " 1529,\n",
       " 1530,\n",
       " 1531,\n",
       " 1534,\n",
       " 1535,\n",
       " 1536,\n",
       " 1537,\n",
       " 1538,\n",
       " 1539,\n",
       " 1540,\n",
       " 1541,\n",
       " 1544,\n",
       " 1545,\n",
       " 1546,\n",
       " 1547,\n",
       " 1548,\n",
       " 1549,\n",
       " 1560,\n",
       " 1561,\n",
       " 1564,\n",
       " 1565,\n",
       " 1566,\n",
       " 1567,\n",
       " 1568,\n",
       " 1569,\n",
       " 1570,\n",
       " 1571,\n",
       " 1574,\n",
       " 1575,\n",
       " 1576,\n",
       " 1577,\n",
       " 1578,\n",
       " 1579,\n",
       " 1580,\n",
       " 1581,\n",
       " 1584,\n",
       " 1585,\n",
       " 1586,\n",
       " 1587,\n",
       " 1588,\n",
       " 1589,\n",
       " 1590,\n",
       " 1591,\n",
       " 1594,\n",
       " 1595,\n",
       " 1596,\n",
       " 1597,\n",
       " 1598,\n",
       " 1599,\n",
       " 1600,\n",
       " 1601,\n",
       " 1604,\n",
       " 1605,\n",
       " 1606,\n",
       " 1607,\n",
       " 1608,\n",
       " 1609,\n",
       " 1620,\n",
       " 1621,\n",
       " 1624,\n",
       " 1625,\n",
       " 1626,\n",
       " 1627,\n",
       " 1628,\n",
       " 1629,\n",
       " 1630,\n",
       " 1631,\n",
       " 1634,\n",
       " 1635,\n",
       " 1636,\n",
       " 1637,\n",
       " 1638,\n",
       " 1639,\n",
       " 1640,\n",
       " 1641,\n",
       " 1644,\n",
       " 1645,\n",
       " 1646,\n",
       " 1647,\n",
       " 1648,\n",
       " 1649,\n",
       " 1660,\n",
       " 1661,\n",
       " 1664,\n",
       " 1665,\n",
       " 1666,\n",
       " 1667,\n",
       " 1668,\n",
       " 1669,\n",
       " 1670,\n",
       " 1671,\n",
       " 1674,\n",
       " 1675,\n",
       " 1676,\n",
       " 1677,\n",
       " 1678,\n",
       " 1679,\n",
       " 1680,\n",
       " 1681,\n",
       " 1684,\n",
       " 1685,\n",
       " 1686,\n",
       " 1687,\n",
       " 1688,\n",
       " 1689,\n",
       " 1690,\n",
       " 1691,\n",
       " 1694,\n",
       " 1695,\n",
       " 1696,\n",
       " 1697,\n",
       " 1698,\n",
       " 1699,\n",
       " 1700,\n",
       " 1701,\n",
       " 1704,\n",
       " 1705,\n",
       " 1706,\n",
       " 1707,\n",
       " 1708,\n",
       " 1709,\n",
       " 1720,\n",
       " 1721,\n",
       " 1724,\n",
       " 1725,\n",
       " 1726,\n",
       " 1727,\n",
       " 1728,\n",
       " 1729,\n",
       " 1730,\n",
       " 1731,\n",
       " 1734,\n",
       " 1735,\n",
       " 1736,\n",
       " 1737,\n",
       " 1738,\n",
       " 1739,\n",
       " 1740,\n",
       " 1741,\n",
       " 1744,\n",
       " 1745,\n",
       " 1746,\n",
       " 1747,\n",
       " 1748,\n",
       " 1749,\n",
       " 1760,\n",
       " 1761,\n",
       " 1764,\n",
       " 1765,\n",
       " 1766,\n",
       " 1767,\n",
       " 1768,\n",
       " 1769,\n",
       " 1770,\n",
       " 1771,\n",
       " 1774,\n",
       " 1775,\n",
       " 1776,\n",
       " 1777,\n",
       " 1778,\n",
       " 1779,\n",
       " 1780,\n",
       " 1781,\n",
       " 1784,\n",
       " 1785,\n",
       " 1786,\n",
       " 1787,\n",
       " 1788,\n",
       " 1789,\n",
       " 1790,\n",
       " 1791,\n",
       " 1794,\n",
       " 1795,\n",
       " 1796,\n",
       " 1797,\n",
       " 1798,\n",
       " 1799,\n",
       " 1800,\n",
       " 1801,\n",
       " 1804,\n",
       " 1805,\n",
       " 1806,\n",
       " 1807,\n",
       " 1808,\n",
       " 1809,\n",
       " 1820,\n",
       " 1821,\n",
       " 1824,\n",
       " 1825,\n",
       " 1826,\n",
       " 1827,\n",
       " 1828,\n",
       " 1829,\n",
       " 1830,\n",
       " 1831,\n",
       " 1834,\n",
       " 1835,\n",
       " 1836,\n",
       " 1837,\n",
       " 1838,\n",
       " 1839,\n",
       " 1840,\n",
       " 1841,\n",
       " 1844,\n",
       " 1845,\n",
       " 1846,\n",
       " 1847,\n",
       " 1848,\n",
       " 1849,\n",
       " 1860,\n",
       " 1861,\n",
       " 1864,\n",
       " 1865,\n",
       " 1866,\n",
       " 1867,\n",
       " 1868,\n",
       " 1869,\n",
       " 1870,\n",
       " 1871,\n",
       " 1874,\n",
       " 1875,\n",
       " 1876,\n",
       " 1877,\n",
       " 1878,\n",
       " 1879,\n",
       " 1880,\n",
       " 1881,\n",
       " 1884,\n",
       " 1885,\n",
       " 1886,\n",
       " 1887,\n",
       " 1888,\n",
       " 1889,\n",
       " 1890,\n",
       " 1891,\n",
       " 1894,\n",
       " 1895,\n",
       " 1896,\n",
       " 1897,\n",
       " 1898,\n",
       " 1899,\n",
       " 1900,\n",
       " 1901,\n",
       " 1904,\n",
       " 1905,\n",
       " 1906,\n",
       " 1907,\n",
       " 1908,\n",
       " 1909,\n",
       " 1920,\n",
       " 1921,\n",
       " 1924,\n",
       " 1925,\n",
       " 1926,\n",
       " 1927,\n",
       " 1928,\n",
       " 1929,\n",
       " 1930,\n",
       " 1931,\n",
       " 1934,\n",
       " 1935,\n",
       " 1936,\n",
       " 1937,\n",
       " 1938,\n",
       " 1939,\n",
       " 1940,\n",
       " 1941,\n",
       " 1944,\n",
       " 1945,\n",
       " 1946,\n",
       " 1947,\n",
       " 1948,\n",
       " 1949,\n",
       " 1960,\n",
       " 1961,\n",
       " 1964,\n",
       " 1965,\n",
       " 1966,\n",
       " 1967,\n",
       " 1968,\n",
       " 1969,\n",
       " 1970,\n",
       " 1971,\n",
       " 1974,\n",
       " 1975,\n",
       " 1976,\n",
       " 1977,\n",
       " 1978,\n",
       " 1979,\n",
       " 1980,\n",
       " 1981,\n",
       " 1984,\n",
       " 1985,\n",
       " 1986,\n",
       " 1987,\n",
       " 1988,\n",
       " 1989,\n",
       " 1990,\n",
       " 1991,\n",
       " 1994,\n",
       " 1995,\n",
       " 1996,\n",
       " 1997,\n",
       " 1998,\n",
       " 1999,\n",
       " 2000,\n",
       " 2001,\n",
       " 2004,\n",
       " 2005,\n",
       " 2006,\n",
       " 2007,\n",
       " 2008,\n",
       " 2009,\n",
       " 2020,\n",
       " 2021,\n",
       " 2024,\n",
       " 2025,\n",
       " 2026,\n",
       " 2027,\n",
       " 2028,\n",
       " 2029,\n",
       " 2030,\n",
       " 2031,\n",
       " 2034,\n",
       " 2035,\n",
       " 2036,\n",
       " 2037,\n",
       " 2038,\n",
       " 2039,\n",
       " 2040,\n",
       " 2041,\n",
       " 2044,\n",
       " 2045,\n",
       " 2046,\n",
       " 2047,\n",
       " 2048,\n",
       " 2049,\n",
       " 2060,\n",
       " 2061,\n",
       " 2064,\n",
       " 2065,\n",
       " 2066,\n",
       " 2067,\n",
       " 2068,\n",
       " 2069,\n",
       " 2070,\n",
       " 2071,\n",
       " 2074,\n",
       " 2075,\n",
       " 2076,\n",
       " 2077,\n",
       " 2078,\n",
       " 2079,\n",
       " 2080,\n",
       " 2081,\n",
       " 2084,\n",
       " 2085,\n",
       " 2086,\n",
       " 2087,\n",
       " 2088,\n",
       " 2089,\n",
       " 2090,\n",
       " 2091,\n",
       " 2094,\n",
       " 2095,\n",
       " 2096,\n",
       " 2097,\n",
       " 2098,\n",
       " 2099,\n",
       " 2100,\n",
       " 2101,\n",
       " 2104,\n",
       " 2105,\n",
       " 2106,\n",
       " 2107,\n",
       " 2108,\n",
       " 2109,\n",
       " 2120,\n",
       " 2121,\n",
       " 2124,\n",
       " 2125,\n",
       " 2126,\n",
       " 2127,\n",
       " 2128,\n",
       " 2129,\n",
       " 2130,\n",
       " 2131,\n",
       " 2134,\n",
       " 2135,\n",
       " 2136,\n",
       " 2137,\n",
       " 2138,\n",
       " 2139,\n",
       " 2140,\n",
       " 2141,\n",
       " 2144,\n",
       " 2145,\n",
       " 2146,\n",
       " 2147,\n",
       " 2148,\n",
       " 2149,\n",
       " 2160,\n",
       " 2161,\n",
       " 2164,\n",
       " 2165,\n",
       " 2166,\n",
       " 2167,\n",
       " 2168,\n",
       " 2169,\n",
       " 2170,\n",
       " 2171,\n",
       " 2174,\n",
       " 2175,\n",
       " 2176,\n",
       " 2177,\n",
       " 2178,\n",
       " 2179,\n",
       " 2180,\n",
       " 2181,\n",
       " 2184,\n",
       " 2185,\n",
       " 2186,\n",
       " 2187,\n",
       " 2188,\n",
       " 2189,\n",
       " 2190,\n",
       " 2191,\n",
       " 2194,\n",
       " 2195,\n",
       " 2196,\n",
       " 2197,\n",
       " 2198,\n",
       " 2199,\n",
       " 2200,\n",
       " 2201,\n",
       " 2204,\n",
       " 2205,\n",
       " 2206,\n",
       " 2207,\n",
       " 2208,\n",
       " 2209,\n",
       " 2220,\n",
       " 2221,\n",
       " 2224,\n",
       " 2225,\n",
       " 2226,\n",
       " 2227,\n",
       " 2228,\n",
       " 2229,\n",
       " 2230,\n",
       " 2231,\n",
       " 2234,\n",
       " 2235,\n",
       " 2236,\n",
       " 2237,\n",
       " 2238,\n",
       " 2239,\n",
       " 2240,\n",
       " 2241,\n",
       " 2244,\n",
       " 2245,\n",
       " 2246,\n",
       " 2247,\n",
       " 2248,\n",
       " 2249,\n",
       " 2260,\n",
       " 2261,\n",
       " 2264,\n",
       " 2265,\n",
       " 2266,\n",
       " 2267,\n",
       " 2268,\n",
       " 2269,\n",
       " 2270,\n",
       " 2271,\n",
       " 2274,\n",
       " 2275,\n",
       " 2276,\n",
       " 2277,\n",
       " 2278,\n",
       " 2279,\n",
       " 2280,\n",
       " 2281,\n",
       " 2284,\n",
       " 2285,\n",
       " 2286,\n",
       " 2287,\n",
       " 2288,\n",
       " 2289,\n",
       " 2290,\n",
       " 2291,\n",
       " 2294,\n",
       " 2295,\n",
       " 2296,\n",
       " 2297,\n",
       " 2298,\n",
       " 2299,\n",
       " 2300,\n",
       " 2301,\n",
       " 2304,\n",
       " 2305,\n",
       " 2306,\n",
       " 2307,\n",
       " 2308,\n",
       " 2309,\n",
       " 2320,\n",
       " 2321,\n",
       " 2324,\n",
       " 2325,\n",
       " 2326,\n",
       " 2327,\n",
       " 2328,\n",
       " 2329,\n",
       " 2330,\n",
       " 2331,\n",
       " 2334,\n",
       " 2335,\n",
       " 2336,\n",
       " 2337,\n",
       " 2338,\n",
       " 2339,\n",
       " 2340,\n",
       " 2341,\n",
       " 2344,\n",
       " 2345,\n",
       " 2346,\n",
       " 2347,\n",
       " 2348,\n",
       " 2349,\n",
       " 2360,\n",
       " 2361,\n",
       " 2364,\n",
       " 2365,\n",
       " 2366,\n",
       " 2367,\n",
       " 2368,\n",
       " 2369,\n",
       " 2370,\n",
       " 2371,\n",
       " 2374,\n",
       " 2375,\n",
       " 2376,\n",
       " 2377,\n",
       " 2378,\n",
       " 2379,\n",
       " 2380,\n",
       " 2381,\n",
       " 2384,\n",
       " 2385,\n",
       " 2386,\n",
       " 2387,\n",
       " 2388,\n",
       " 2389,\n",
       " 2390,\n",
       " 2391,\n",
       " 2394,\n",
       " 2395,\n",
       " 2396,\n",
       " 2397,\n",
       " 2398,\n",
       " 2399,\n",
       " 2500,\n",
       " 2501,\n",
       " 2504,\n",
       " 2505,\n",
       " 2506,\n",
       " 2507,\n",
       " 2508,\n",
       " 2509,\n",
       " 2520,\n",
       " 2521,\n",
       " 2524,\n",
       " 2525,\n",
       " 2526,\n",
       " 2527,\n",
       " 2528,\n",
       " 2529,\n",
       " 2530,\n",
       " 2531,\n",
       " 2534,\n",
       " 2535,\n",
       " 2536,\n",
       " 2537,\n",
       " 2538,\n",
       " 2539,\n",
       " 2540,\n",
       " 2541,\n",
       " 2544,\n",
       " 2545,\n",
       " 2546,\n",
       " 2547,\n",
       " 2548,\n",
       " 2549,\n",
       " 2560,\n",
       " 2561,\n",
       " 2564,\n",
       " 2565,\n",
       " 2566,\n",
       " 2567,\n",
       " 2568,\n",
       " 2569,\n",
       " 2570,\n",
       " 2571,\n",
       " 2574,\n",
       " 2575,\n",
       " 2576,\n",
       " 2577,\n",
       " 2578,\n",
       " 2579,\n",
       " 2580,\n",
       " 2581,\n",
       " 2584,\n",
       " 2585,\n",
       " 2586,\n",
       " 2587,\n",
       " 2588,\n",
       " 2589,\n",
       " 2590,\n",
       " 2591,\n",
       " 2594,\n",
       " 2595,\n",
       " 2596,\n",
       " 2597,\n",
       " 2598,\n",
       " 2599,\n",
       " 2600,\n",
       " 2601,\n",
       " 2604,\n",
       " 2605,\n",
       " 2606,\n",
       " 2607,\n",
       " 2608,\n",
       " 2609,\n",
       " 2620,\n",
       " 2621,\n",
       " 2624,\n",
       " 2625,\n",
       " 2626,\n",
       " 2627,\n",
       " 2628,\n",
       " 2629,\n",
       " 2630,\n",
       " 2631,\n",
       " 2634,\n",
       " 2635,\n",
       " 2636,\n",
       " 2637,\n",
       " 2638,\n",
       " 2639,\n",
       " 2640,\n",
       " 2641,\n",
       " 2644,\n",
       " 2645,\n",
       " 2646,\n",
       " 2647,\n",
       " 2648,\n",
       " 2649,\n",
       " 2660,\n",
       " 2661,\n",
       " 2664,\n",
       " 2665,\n",
       " 2666,\n",
       " 2667,\n",
       " 2668,\n",
       " 2669,\n",
       " 2670,\n",
       " 2671,\n",
       " 2674,\n",
       " 2675,\n",
       " 2676,\n",
       " 2677,\n",
       " 2678,\n",
       " 2679,\n",
       " 2680,\n",
       " 2681,\n",
       " 2684,\n",
       " 2685,\n",
       " 2686,\n",
       " 2687,\n",
       " 2688,\n",
       " 2689,\n",
       " 2690,\n",
       " 2691,\n",
       " 2694,\n",
       " 2695,\n",
       " 2696,\n",
       " 2697,\n",
       " 2698,\n",
       " 2699,\n",
       " 2700,\n",
       " 2701,\n",
       " 2704,\n",
       " 2705,\n",
       " 2706,\n",
       " 2707,\n",
       " 2708,\n",
       " 2709,\n",
       " 2720,\n",
       " 2721,\n",
       " 2724,\n",
       " 2725,\n",
       " 2726,\n",
       " 2727,\n",
       " 2728,\n",
       " 2729,\n",
       " 2730,\n",
       " 2731,\n",
       " 2734,\n",
       " 2735,\n",
       " 2736,\n",
       " 2737,\n",
       " 2738,\n",
       " 2739,\n",
       " 2740,\n",
       " 2741,\n",
       " 2744,\n",
       " 2745,\n",
       " 2746,\n",
       " 2747,\n",
       " 2748,\n",
       " 2749,\n",
       " 2760,\n",
       " 2761,\n",
       " 2764,\n",
       " 2765,\n",
       " 2766,\n",
       " 2767,\n",
       " 2768,\n",
       " 2769,\n",
       " ...}"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{\n",
    "    c1 * 1000 + c2 * 100 + c3 * 10 + c4\n",
    "    for c4 in {0,1,4,5,6,7,8,9} \n",
    "    for c2 in {0,1,2,3,5,6,7,8,9}\n",
    "    for c3 in {0,2,3,4,6,7,8,9}\n",
    "    for c1 in {1,2,3,4,5,6,7,8,9}\n",
    "}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5184"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "9 * 8 * 9 * 8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$A$ und $B$ heißen disjunkt, wenn $A \\cap B = \\emptyset$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 5. $\\mathbb{N+}$ Zahlen <= 1000, nicht durch 2, 3, 5 teilbar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "266"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len({x for x in range(1001) if x % 2 != 0 and x % 3 != 0 and x % 5 != 0})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 6. Mathematisches summieren mit Sigma ($\\Sigma$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(5050, 338350)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# s muss eine funktion sein mit genau einem parameter\n",
    "def newsum(s, a, b):\n",
    "    result = 0\n",
    "    counter = a\n",
    "    \n",
    "    while counter <= b:\n",
    "        result += s(counter) # funktion s mit einem parameter\n",
    "        counter += 1\n",
    "\n",
    "    return result\n",
    "\n",
    "# beispiel für s:\n",
    "# identische abbildung (keine veränderung)\n",
    "def id(x):\n",
    "    return x\n",
    "\n",
    "# beispiel für s:\n",
    "# quadrieren (y*y)\n",
    "def square(y):\n",
    "    return y**2\n",
    "\n",
    "newsum(id, 1, 100), newsum(square, 1, 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 7. Gaußsche Summenformel"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\sum_{i=1}^{n} i = \\frac{n(n+1)}{2}$\n",
    "\n",
    "für gerade $n$: $\\frac{n}{2}(n+1)$\n",
    "\n",
    "für ungerade $n$: $\\frac{n-1}{2}(n+1)+(\\frac{n+1}{2}) = \\frac{n(n+1)}{2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(55, 55.0, 66, 66.0, 5050, 5050.0)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Iterativ, rechenaufwändig\n",
    "def gaussian_sum_expensive(n):\n",
    "    result = 0\n",
    "    for i in range(n + 1):\n",
    "        result += i\n",
    "    return result\n",
    "\n",
    "# Gekürzt, weniger rechenaufwändig\n",
    "def gaussian_sum(n):\n",
    "    if n % 2 == 0:\n",
    "        return (n/2)*(n+1)\n",
    "    else:\n",
    "        return n*(n+1)/2\n",
    "\n",
    "(\n",
    "    gaussian_sum_expensive(10), gaussian_sum(10), # 10\n",
    "    gaussian_sum_expensive(11), gaussian_sum(11), # 11\n",
    "    gaussian_sum_expensive(100), gaussian_sum(100) # 100\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 8. Faulhabersche Formel\n",
    "\n",
    "$\\sum_{k=1}^{n} k^{2} = \\frac{1}{3}n^{3} + \\frac{1}{2}n^{2} + \\frac{1}{6}n$\n",
    "\n",
    "$= 3*1 + 3*2 + 3*3 + 3*4 + 3*5$\n",
    "\n",
    "$= 3 (1+2+3+4+5)$\n",
    "\n",
    "$= 3 \\sum_{k=1}^{5} k$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 9. Summieren von $a$ bis $b$ (statt $1$ bis $b$)\n",
    "\n",
    "$\\sum_{k=10}^{100} k = \\sum_{k=1}^{100} k - \\sum_{k=1}^{9} k$\n",
    "\n",
    "$\\sum_{k=a}^{b} S_{k} = \\sum_{k=a}^{b} S_{k} - \\sum_{k=a}^{a-1} S_{k}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(45.0, 3825.0, 42.0)"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def range_sum(a, b):\n",
    "    sum_big = gaussian_sum(b)\n",
    "    sum_small = gaussian_sum(a - 1)\n",
    "    \n",
    "    return sum_big - sum_small\n",
    "\n",
    "range_sum(5, 10), range_sum(50, 100), range_sum(3, 9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 10 . Summe von zwei Summenfunktionen\n",
    "\n",
    "$\\sum_{k=a}^{b} (U_{k}+V_{k}) = \\sum_{k=a}^{b} U_{k} + \\sum_{k=a}^{b} V_{k}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 11. Geometrische Summenformel\n",
    "\n",
    "$\\sum_{i=0}^{5} 3^{i} = 3^{0} + 3^{1} + 3^{2} + 3^{3} + 3^{4} + 3^{5} = S$\n",
    "\n",
    "$3 * S = 3^{1} + 3^{2} + 3^{3} + 3^{4} + 3^{5}+ 3^{6}$\n",
    "\n",
    "$S = \\frac{1-3^{6}}{1-3}$\n",
    "\n",
    "$\\sum_{i=0}^{n} q^{i} = \\frac{1-q^{n+1}}{1-q}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(121, 121.0)"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Rechenaufwändig, manuell\n",
    "def geometric_sum_expensive(n, q, i):\n",
    "    result = 0\n",
    "    for i in range(n + 1):\n",
    "        result += q ** i\n",
    "    return result\n",
    "\n",
    "# Gekürzte variante, weniger rechenaufwändig\n",
    "def geometric_sum(n, q, i):\n",
    "    return (1 - (q ** (n + 1))) / (1 - q)\n",
    "\n",
    "geometric_sum_expensive(4, 3, 2), geometric_sum(4, 3, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 12. Das Produktzeichen\n",
    "\n",
    "$U_{1} * U_{2} = \\prod_{i=1}^{2} U_{i}$\n",
    "\n",
    "$\\prod_{i=1}^{n} i = 1*2*3*...*n$ (Fakultät)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, 120, 3628800)"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def facul(n):\n",
    "    return n * facul(n - 1) if n > 1 else 1\n",
    "\n",
    "facul(1), facul(5), facul(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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