[{"data":1,"prerenderedAt":3146},["ShallowReactive",2],{"subject-categories-math":3,"formula-de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fmaechtigkeit-der-potenzmenge":546,"formula-related-de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fmaechtigkeit-der-potenzmenge":3136},[4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,173,174,175,176,177,178,179,180,181,182,183,184,185,186,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545],{"category":5},"algebra",{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":65},"arithmetic",{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":101},"calculus",{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":172},"differential-equations",{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":187},"discrete-math",{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":208},"functions",{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":247},"geometry",{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":312},"linear-algebra",{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":368},"multivariable-calculus",{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":391},"numerics",{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":412},"sequences-series",{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":441},"stochastics",{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":518},"trigonometry",{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"id":547,"title":548,"aliases":549,"body":555,"category":187,"checks":3083,"conditions":3099,"conditionsHtml":3100,"deprecated":3101,"description":3077,"extension":3102,"figure":3101,"figureCaption":3101,"figureHtml":3101,"formulaId":3103,"latex":3104,"latexHtml":3105,"level":3106,"meta":3107,"navigation":3108,"path":3109,"redirectFrom":3110,"related":3111,"seo":3115,"stem":3116,"subject":582,"summary":3117,"tags":3118,"variables":3124,"__hash__":3135},"formulas_de\u002Fde\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fmaechtigkeit-der-potenzmenge.md","Mächtigkeit der Potenzmenge",[550,551,552,553,554],"Anzahl der Teilmengen","Potenzmenge Anzahl Elemente","2 hoch n Teilmengen","Kardinalität der Potenzmenge","wie viele Teilmengen hat eine Menge",{"type":556,"value":557,"toc":3076},"minimark",[558,563,916,1136,1171,1438,1505,1509,1691,1986,2109,2228,2415,2419],[559,560,562],"h2",{"id":561},"erklärung","Erklärung",[564,565,566,567,571,572,645,646,675,676,708,709,737,738,766,767,797,798,766,841,915],"p",{},"Die ",[568,569,570],"strong",{},"Potenzmenge"," ",[573,574,577,612],"span",{"className":575},[576],"katex",[573,578,581],{"className":579},[580],"katex-mathml",[582,583,585],"math",{"xmlns":584},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[586,587,588,607],"semantics",{},[589,590,591,596,601,604],"mrow",{},[592,593,595],"mi",{"mathvariant":594},"script","P",[597,598,600],"mo",{"stretchy":599},"false","(",[592,602,603],{},"M",[597,605,606],{"stretchy":599},")",[608,609,611],"annotation",{"encoding":610},"application\u002Fx-tex","\\mathcal{P}(M)",[573,613,617],{"className":614,"ariaHidden":616},[615],"katex-html","true",[573,618,621,626,632,636,641],{"className":619},[620],"base",[573,622],{"className":623,"style":625},[624],"strut","height:1em;vertical-align:-0.25em;",[573,627,595],{"className":628,"style":631},[629,630],"mord","mathcal","margin-right:0.0822em;",[573,633,600],{"className":634},[635],"mopen",[573,637,603],{"className":638,"style":640},[629,639],"mathnormal","margin-right:0.109em;",[573,642,606],{"className":643},[644],"mclose"," enthält alle Teilmengen von ",[573,647,649,662],{"className":648},[576],[573,650,652],{"className":651},[580],[582,653,654],{"xmlns":584},[586,655,656,660],{},[589,657,658],{},[592,659,603],{},[608,661,603],{"encoding":610},[573,663,665],{"className":664,"ariaHidden":616},[615],[573,666,668,672],{"className":667},[620],[573,669],{"className":670,"style":671},[624],"height:0.6833em;",[573,673,603],{"className":674,"style":640},[629,639],", auch die leere Menge ",[573,677,679,695],{"className":678},[576],[573,680,682],{"className":681},[580],[582,683,684],{"xmlns":584},[586,685,686,692],{},[589,687,688],{},[592,689,691],{"mathvariant":690},"normal","∅",[608,693,694],{"encoding":610},"\\emptyset",[573,696,698],{"className":697,"ariaHidden":616},[615],[573,699,701,705],{"className":700},[620],[573,702],{"className":703,"style":704},[624],"height:0.8056em;vertical-align:-0.0556em;",[573,706,691],{"className":707},[629]," und ",[573,710,712,725],{"className":711},[576],[573,713,715],{"className":714},[580],[582,716,717],{"xmlns":584},[586,718,719,723],{},[589,720,721],{},[592,722,603],{},[608,724,603],{"encoding":610},[573,726,728],{"className":727,"ariaHidden":616},[615],[573,729,731,734],{"className":730},[620],[573,732],{"className":733,"style":671},[624],[573,735,603],{"className":736,"style":640},[629,639]," selbst. Hat ",[573,739,741,754],{"className":740},[576],[573,742,744],{"className":743},[580],[582,745,746],{"xmlns":584},[586,747,748,752],{},[589,749,750],{},[592,751,603],{},[608,753,603],{"encoding":610},[573,755,757],{"className":756,"ariaHidden":616},[615],[573,758,760,763],{"className":759},[620],[573,761],{"className":762,"style":671},[624],[573,764,603],{"className":765,"style":640},[629,639]," genau ",[573,768,770,784],{"className":769},[576],[573,771,773],{"className":772},[580],[582,774,775],{"xmlns":584},[586,776,777,782],{},[589,778,779],{},[592,780,781],{},"n",[608,783,781],{"encoding":610},[573,785,787],{"className":786,"ariaHidden":616},[615],[573,788,790,794],{"className":789},[620],[573,791],{"className":792,"style":793},[624],"height:0.4306em;",[573,795,781],{"className":796},[629,639]," Elemente, so hat ",[573,799,801,820],{"className":800},[576],[573,802,804],{"className":803},[580],[582,805,806],{"xmlns":584},[586,807,808,818],{},[589,809,810,812,814,816],{},[592,811,595],{"mathvariant":594},[597,813,600],{"stretchy":599},[592,815,603],{},[597,817,606],{"stretchy":599},[608,819,611],{"encoding":610},[573,821,823],{"className":822,"ariaHidden":616},[615],[573,824,826,829,832,835,838],{"className":825},[620],[573,827],{"className":828,"style":625},[624],[573,830,595],{"className":831,"style":631},[629,630],[573,833,600],{"className":834},[635],[573,836,603],{"className":837,"style":640},[629,639],[573,839,606],{"className":840},[644],[573,842,844,865],{"className":843},[576],[573,845,847],{"className":846},[580],[582,848,849],{"xmlns":584},[586,850,851,862],{},[589,852,853],{},[854,855,856,860],"msup",{},[857,858,859],"mn",{},"2",[592,861,781],{},[608,863,864],{"encoding":610},"2^n",[573,866,868],{"className":867,"ariaHidden":616},[615],[573,869,871,875],{"className":870},[620],[573,872],{"className":873,"style":874},[624],"height:0.6644em;",[573,876,878,881],{"className":877},[629],[573,879,859],{"className":880},[629],[573,882,885],{"className":883},[884],"msupsub",[573,886,889],{"className":887},[888],"vlist-t",[573,890,893],{"className":891},[892],"vlist-r",[573,894,897],{"className":895,"style":874},[896],"vlist",[573,898,900,905],{"style":899},"top:-3.063em;margin-right:0.05em;",[573,901],{"className":902,"style":904},[903],"pstrut","height:2.7em;",[573,906,912],{"className":907},[908,909,910,911],"sizing","reset-size6","size3","mtight",[573,913,781],{"className":914},[629,639,911]," Elemente.",[564,917,918,919,947,948,1106,1107,1135],{},"Begründung: Um eine Teilmenge festzulegen, entscheidet man für jedes der ",[573,920,922,935],{"className":921},[576],[573,923,925],{"className":924},[580],[582,926,927],{"xmlns":584},[586,928,929,933],{},[589,930,931],{},[592,932,781],{},[608,934,781],{"encoding":610},[573,936,938],{"className":937,"ariaHidden":616},[615],[573,939,941,944],{"className":940},[620],[573,942],{"className":943,"style":793},[624],[573,945,781],{"className":946},[629,639]," Elemente unabhängig, ob es dazugehört oder nicht. Das sind ",[573,949,951,988],{"className":950},[576],[573,952,954],{"className":953},[580],[582,955,956],{"xmlns":584},[586,957,958,985],{},[589,959,960,962,965,967,969,972,974,976,979],{},[857,961,859],{},[597,963,964],{},"⋅",[857,966,859],{},[597,968,964],{},[597,970,971],{},"…",[597,973,964],{},[857,975,859],{},[597,977,978],{},"=",[854,980,981,983],{},[857,982,859],{},[592,984,781],{},[608,986,987],{"encoding":610},"2 \\cdot 2 \\cdot \\ldots \\cdot 2 = 2^n",[573,989,991,1013,1031,1051,1071],{"className":990,"ariaHidden":616},[615],[573,992,994,998,1001,1006,1010],{"className":993},[620],[573,995],{"className":996,"style":997},[624],"height:0.6444em;",[573,999,859],{"className":1000},[629],[573,1002],{"className":1003,"style":1005},[1004],"mspace","margin-right:0.2222em;",[573,1007,964],{"className":1008},[1009],"mbin",[573,1011],{"className":1012,"style":1005},[1004],[573,1014,1016,1019,1022,1025,1028],{"className":1015},[620],[573,1017],{"className":1018,"style":997},[624],[573,1020,859],{"className":1021},[629],[573,1023],{"className":1024,"style":1005},[1004],[573,1026,964],{"className":1027},[1009],[573,1029],{"className":1030,"style":1005},[1004],[573,1032,1034,1038,1042,1045,1048],{"className":1033},[620],[573,1035],{"className":1036,"style":1037},[624],"height:0.4445em;",[573,1039,971],{"className":1040},[1041],"minner",[573,1043],{"className":1044,"style":1005},[1004],[573,1046,964],{"className":1047},[1009],[573,1049],{"className":1050,"style":1005},[1004],[573,1052,1054,1057,1060,1064,1068],{"className":1053},[620],[573,1055],{"className":1056,"style":997},[624],[573,1058,859],{"className":1059},[629],[573,1061],{"className":1062,"style":1063},[1004],"margin-right:0.2778em;",[573,1065,978],{"className":1066},[1067],"mrel",[573,1069],{"className":1070,"style":1063},[1004],[573,1072,1074,1077],{"className":1073},[620],[573,1075],{"className":1076,"style":874},[624],[573,1078,1080,1083],{"className":1079},[629],[573,1081,859],{"className":1082},[629],[573,1084,1086],{"className":1085},[884],[573,1087,1089],{"className":1088},[888],[573,1090,1092],{"className":1091},[892],[573,1093,1095],{"className":1094,"style":874},[896],[573,1096,1097,1100],{"style":899},[573,1098],{"className":1099,"style":904},[903],[573,1101,1103],{"className":1102},[908,909,910,911],[573,1104,781],{"className":1105},[629,639,911]," Möglichkeiten. Gleichwertig entspricht jede Teilmenge genau einem 0-1-Wort der Länge ",[573,1108,1110,1123],{"className":1109},[576],[573,1111,1113],{"className":1112},[580],[582,1114,1115],{"xmlns":584},[586,1116,1117,1121],{},[589,1118,1119],{},[592,1120,781],{},[608,1122,781],{"encoding":610},[573,1124,1126],{"className":1125,"ariaHidden":616},[615],[573,1127,1129,1132],{"className":1128},[620],[573,1130],{"className":1131,"style":793},[624],[573,1133,781],{"className":1134},[629,639],".",[564,1137,1138,1139,1170],{},"Zählt man die Teilmengen nach ihrer Größe ",[573,1140,1142,1156],{"className":1141},[576],[573,1143,1145],{"className":1144},[580],[582,1146,1147],{"xmlns":584},[586,1148,1149,1154],{},[589,1150,1151],{},[592,1152,1153],{},"k",[608,1155,1153],{"encoding":610},[573,1157,1159],{"className":1158,"ariaHidden":616},[615],[573,1160,1162,1166],{"className":1161},[620],[573,1163],{"className":1164,"style":1165},[624],"height:0.6944em;",[573,1167,1153],{"className":1168,"style":1169},[629,639],"margin-right:0.0315em;",", erhält man die Zeilensumme im Pascalschen Dreieck:",[573,1172,1175],{"className":1173},[1174],"katex-display",[573,1176,1178,1230],{"className":1177},[576],[573,1179,1181],{"className":1180},[580],[582,1182,1184],{"xmlns":584,"display":1183},"block",[586,1185,1186,1227],{},[589,1187,1188,1205,1219,1221],{},[1189,1190,1191,1194,1203],"munderover",{},[597,1192,1193],{},"∑",[589,1195,1196,1198,1200],{},[592,1197,1153],{},[597,1199,978],{},[857,1201,1202],{},"0",[592,1204,781],{},[589,1206,1207,1209,1217],{},[597,1208,600],{"fence":616},[1210,1211,1213,1215],"mfrac",{"linethickness":1212},"0px",[592,1214,781],{},[592,1216,1153],{},[597,1218,606],{"fence":616},[597,1220,978],{},[854,1222,1223,1225],{},[857,1224,859],{},[592,1226,781],{},[608,1228,1229],{"encoding":610},"\\sum_{k=0}^{n} \\binom{n}{k} = 2^n",[573,1231,1233,1401],{"className":1232,"ariaHidden":616},[615],[573,1234,1236,1240,1320,1324,1392,1395,1398],{"className":1235},[620],[573,1237],{"className":1238,"style":1239},[624],"height:2.9535em;vertical-align:-1.3021em;",[573,1241,1245],{"className":1242},[1243,1244],"mop","op-limits",[573,1246,1249,1311],{"className":1247},[888,1248],"vlist-t2",[573,1250,1252,1306],{"className":1251},[892],[573,1253,1256,1278,1291],{"className":1254,"style":1255},[896],"height:1.6514em;",[573,1257,1259,1263],{"style":1258},"top:-1.8479em;margin-left:0em;",[573,1260],{"className":1261,"style":1262},[903],"height:3.05em;",[573,1264,1266],{"className":1265},[908,909,910,911],[573,1267,1269,1272,1275],{"className":1268},[629,911],[573,1270,1153],{"className":1271,"style":1169},[629,639,911],[573,1273,978],{"className":1274},[1067,911],[573,1276,1202],{"className":1277},[629,911],[573,1279,1281,1284],{"style":1280},"top:-3.05em;",[573,1282],{"className":1283,"style":1262},[903],[573,1285,1286],{},[573,1287,1193],{"className":1288},[1243,1289,1290],"op-symbol","large-op",[573,1292,1294,1297],{"style":1293},"top:-4.3em;margin-left:0em;",[573,1295],{"className":1296,"style":1262},[903],[573,1298,1300],{"className":1299},[908,909,910,911],[573,1301,1303],{"className":1302},[629,911],[573,1304,781],{"className":1305},[629,639,911],[573,1307,1310],{"className":1308},[1309],"vlist-s","​",[573,1312,1314],{"className":1313},[892],[573,1315,1318],{"className":1316,"style":1317},[896],"height:1.3021em;",[573,1319],{},[573,1321],{"className":1322,"style":1323},[1004],"margin-right:0.1667em;",[573,1325,1327,1336,1386],{"className":1326},[629],[573,1328,1332],{"className":1329,"style":1331},[635,1330],"delimcenter","top:0em;",[573,1333,600],{"className":1334},[1335,910],"delimsizing",[573,1337,1339],{"className":1338},[1210],[573,1340,1342,1377],{"className":1341},[888,1248],[573,1343,1345,1374],{"className":1344},[892],[573,1346,1349,1362],{"className":1347,"style":1348},[896],"height:1.1076em;",[573,1350,1352,1356],{"style":1351},"top:-2.314em;",[573,1353],{"className":1354,"style":1355},[903],"height:3em;",[573,1357,1359],{"className":1358},[629],[573,1360,1153],{"className":1361,"style":1169},[629,639],[573,1363,1365,1368],{"style":1364},"top:-3.677em;",[573,1366],{"className":1367,"style":1355},[903],[573,1369,1371],{"className":1370},[629],[573,1372,781],{"className":1373},[629,639],[573,1375,1310],{"className":1376},[1309],[573,1378,1380],{"className":1379},[892],[573,1381,1384],{"className":1382,"style":1383},[896],"height:0.686em;",[573,1385],{},[573,1387,1389],{"className":1388,"style":1331},[644,1330],[573,1390,606],{"className":1391},[1335,910],[573,1393],{"className":1394,"style":1063},[1004],[573,1396,978],{"className":1397},[1067],[573,1399],{"className":1400,"style":1063},[1004],[573,1402,1404,1408],{"className":1403},[620],[573,1405],{"className":1406,"style":1407},[624],"height:0.7144em;",[573,1409,1411,1414],{"className":1410},[629],[573,1412,859],{"className":1413},[629],[573,1415,1417],{"className":1416},[884],[573,1418,1420],{"className":1419},[888],[573,1421,1423],{"className":1422},[892],[573,1424,1426],{"className":1425,"style":1407},[896],[573,1427,1429,1432],{"style":1428},"top:-3.113em;margin-right:0.05em;",[573,1430],{"className":1431,"style":904},[903],[573,1433,1435],{"className":1434},[908,909,910,911],[573,1436,781],{"className":1437},[629,639,911],[564,1439,1440,1441,1444,1445,1135],{},"Für unendliche Mengen gilt der ",[568,1442,1443],{},"Satz von Cantor",": Die Potenzmenge ist immer echt mächtiger als die Menge selbst. Deshalb schreibt man die Potenzmenge auch als ",[573,1446,1448,1466],{"className":1447},[576],[573,1449,1451],{"className":1450},[580],[582,1452,1453],{"xmlns":584},[586,1454,1455,1463],{},[589,1456,1457],{},[854,1458,1459,1461],{},[857,1460,859],{},[592,1462,603],{},[608,1464,1465],{"encoding":610},"2^M",[573,1467,1469],{"className":1468,"ariaHidden":616},[615],[573,1470,1472,1476],{"className":1471},[620],[573,1473],{"className":1474,"style":1475},[624],"height:0.8413em;",[573,1477,1479,1482],{"className":1478},[629],[573,1480,859],{"className":1481},[629],[573,1483,1485],{"className":1484},[884],[573,1486,1488],{"className":1487},[888],[573,1489,1491],{"className":1490},[892],[573,1492,1494],{"className":1493,"style":1475},[896],[573,1495,1496,1499],{"style":899},[573,1497],{"className":1498,"style":904},[903],[573,1500,1502],{"className":1501},[908,909,910,911],[573,1503,603],{"className":1504,"style":640},[629,639,911],[559,1506,1508],{"id":1507},"beispiele","Beispiele",[564,1510,1511,1605,1606,1690],{},[573,1512,1514,1550],{"className":1513},[576],[573,1515,1517],{"className":1516},[580],[582,1518,1519],{"xmlns":584},[586,1520,1521,1547],{},[589,1522,1523,1525,1527,1530,1533,1536,1539,1541,1544],{},[592,1524,603],{},[597,1526,978],{},[597,1528,1529],{"stretchy":599},"{",[592,1531,1532],{},"a",[597,1534,1535],{"separator":616},",",[592,1537,1538],{},"b",[597,1540,1535],{"separator":616},[592,1542,1543],{},"c",[597,1545,1546],{"stretchy":599},"}",[608,1548,1549],{"encoding":610},"M = \\{a, b, c\\}",[573,1551,1553,1571],{"className":1552,"ariaHidden":616},[615],[573,1554,1556,1559,1562,1565,1568],{"className":1555},[620],[573,1557],{"className":1558,"style":671},[624],[573,1560,603],{"className":1561,"style":640},[629,639],[573,1563],{"className":1564,"style":1063},[1004],[573,1566,978],{"className":1567},[1067],[573,1569],{"className":1570,"style":1063},[1004],[573,1572,1574,1577,1580,1583,1587,1590,1593,1596,1599,1602],{"className":1573},[620],[573,1575],{"className":1576,"style":625},[624],[573,1578,1529],{"className":1579},[635],[573,1581,1532],{"className":1582},[629,639],[573,1584,1535],{"className":1585},[1586],"mpunct",[573,1588],{"className":1589,"style":1323},[1004],[573,1591,1538],{"className":1592},[629,639],[573,1594,1535],{"className":1595},[1586],[573,1597],{"className":1598,"style":1323},[1004],[573,1600,1543],{"className":1601},[629,639],[573,1603,1546],{"className":1604},[644]," hat ",[573,1607,1609,1633],{"className":1608},[576],[573,1610,1612],{"className":1611},[580],[582,1613,1614],{"xmlns":584},[586,1615,1616,1630],{},[589,1617,1618,1625,1627],{},[854,1619,1620,1622],{},[857,1621,859],{},[857,1623,1624],{},"3",[597,1626,978],{},[857,1628,1629],{},"8",[608,1631,1632],{"encoding":610},"2^3 = 8",[573,1634,1636,1681],{"className":1635,"ariaHidden":616},[615],[573,1637,1639,1643,1672,1675,1678],{"className":1638},[620],[573,1640],{"className":1641,"style":1642},[624],"height:0.8141em;",[573,1644,1646,1649],{"className":1645},[629],[573,1647,859],{"className":1648},[629],[573,1650,1652],{"className":1651},[884],[573,1653,1655],{"className":1654},[888],[573,1656,1658],{"className":1657},[892],[573,1659,1661],{"className":1660,"style":1642},[896],[573,1662,1663,1666],{"style":899},[573,1664],{"className":1665,"style":904},[903],[573,1667,1669],{"className":1668},[908,909,910,911],[573,1670,1624],{"className":1671},[629,911],[573,1673],{"className":1674,"style":1063},[1004],[573,1676,978],{"className":1677},[1067],[573,1679],{"className":1680,"style":1063},[1004],[573,1682,1684,1687],{"className":1683},[620],[573,1685],{"className":1686,"style":997},[624],[573,1688,1629],{"className":1689},[629]," Teilmengen:",[573,1692,1694],{"className":1693},[1174],[573,1695,1697,1803],{"className":1696},[576],[573,1698,1700],{"className":1699},[580],[582,1701,1702],{"xmlns":584,"display":1183},[586,1703,1704,1800],{},[589,1705,1706,1708,1710,1714,1716,1718,1720,1722,1724,1726,1728,1730,1732,1734,1736,1738,1740,1742,1744,1746,1748,1750,1752,1754,1756,1758,1760,1762,1764,1766,1768,1770,1772,1774,1776,1778,1780,1782,1784,1786,1788,1790,1792,1794,1796,1798],{},[592,1707,691],{"mathvariant":690},[597,1709,1535],{"separator":616},[1711,1712,1713],"mtext",{}," ",[597,1715,1529],{"stretchy":599},[592,1717,1532],{},[597,1719,1546],{"stretchy":599},[597,1721,1535],{"separator":616},[1711,1723,1713],{},[597,1725,1529],{"stretchy":599},[592,1727,1538],{},[597,1729,1546],{"stretchy":599},[597,1731,1535],{"separator":616},[1711,1733,1713],{},[597,1735,1529],{"stretchy":599},[592,1737,1543],{},[597,1739,1546],{"stretchy":599},[597,1741,1535],{"separator":616},[1711,1743,1713],{},[597,1745,1529],{"stretchy":599},[592,1747,1532],{},[597,1749,1535],{"separator":616},[592,1751,1538],{},[597,1753,1546],{"stretchy":599},[597,1755,1535],{"separator":616},[1711,1757,1713],{},[597,1759,1529],{"stretchy":599},[592,1761,1532],{},[597,1763,1535],{"separator":616},[592,1765,1543],{},[597,1767,1546],{"stretchy":599},[597,1769,1535],{"separator":616},[1711,1771,1713],{},[597,1773,1529],{"stretchy":599},[592,1775,1538],{},[597,1777,1535],{"separator":616},[592,1779,1543],{},[597,1781,1546],{"stretchy":599},[597,1783,1535],{"separator":616},[1711,1785,1713],{},[597,1787,1529],{"stretchy":599},[592,1789,1532],{},[597,1791,1535],{"separator":616},[592,1793,1538],{},[597,1795,1535],{"separator":616},[592,1797,1543],{},[597,1799,1546],{"stretchy":599},[608,1801,1802],{"encoding":610},"\\emptyset,\\ \\{a\\},\\ \\{b\\},\\ \\{c\\},\\ \\{a, b\\},\\ \\{a, c\\},\\ \\{b, c\\},\\ \\{a, b, c\\}",[573,1804,1806],{"className":1805,"ariaHidden":616},[615],[573,1807,1809,1812,1815,1818,1821,1824,1827,1830,1833,1836,1839,1842,1845,1848,1851,1854,1857,1860,1863,1866,1869,1872,1875,1878,1881,1884,1887,1890,1893,1896,1899,1902,1905,1908,1911,1914,1917,1920,1923,1926,1929,1932,1935,1938,1941,1944,1947,1950,1953,1956,1959,1962,1965,1968,1971,1974,1977,1980,1983],{"className":1808},[620],[573,1810],{"className":1811,"style":625},[624],[573,1813,691],{"className":1814},[629],[573,1816,1535],{"className":1817},[1586],[573,1819,1713],{"className":1820},[1004],[573,1822],{"className":1823,"style":1323},[1004],[573,1825,1529],{"className":1826},[635],[573,1828,1532],{"className":1829},[629,639],[573,1831,1546],{"className":1832},[644],[573,1834,1535],{"className":1835},[1586],[573,1837,1713],{"className":1838},[1004],[573,1840],{"className":1841,"style":1323},[1004],[573,1843,1529],{"className":1844},[635],[573,1846,1538],{"className":1847},[629,639],[573,1849,1546],{"className":1850},[644],[573,1852,1535],{"className":1853},[1586],[573,1855,1713],{"className":1856},[1004],[573,1858],{"className":1859,"style":1323},[1004],[573,1861,1529],{"className":1862},[635],[573,1864,1543],{"className":1865},[629,639],[573,1867,1546],{"className":1868},[644],[573,1870,1535],{"className":1871},[1586],[573,1873,1713],{"className":1874},[1004],[573,1876],{"className":1877,"style":1323},[1004],[573,1879,1529],{"className":1880},[635],[573,1882,1532],{"className":1883},[629,639],[573,1885,1535],{"className":1886},[1586],[573,1888],{"className":1889,"style":1323},[1004],[573,1891,1538],{"className":1892},[629,639],[573,1894,1546],{"className":1895},[644],[573,1897,1535],{"className":1898},[1586],[573,1900,1713],{"className":1901},[1004],[573,1903],{"className":1904,"style":1323},[1004],[573,1906,1529],{"className":1907},[635],[573,1909,1532],{"className":1910},[629,639],[573,1912,1535],{"className":1913},[1586],[573,1915],{"className":1916,"style":1323},[1004],[573,1918,1543],{"className":1919},[629,639],[573,1921,1546],{"className":1922},[644],[573,1924,1535],{"className":1925},[1586],[573,1927,1713],{"className":1928},[1004],[573,1930],{"className":1931,"style":1323},[1004],[573,1933,1529],{"className":1934},[635],[573,1936,1538],{"className":1937},[629,639],[573,1939,1535],{"className":1940},[1586],[573,1942],{"className":1943,"style":1323},[1004],[573,1945,1543],{"className":1946},[629,639],[573,1948,1546],{"className":1949},[644],[573,1951,1535],{"className":1952},[1586],[573,1954,1713],{"className":1955},[1004],[573,1957],{"className":1958,"style":1323},[1004],[573,1960,1529],{"className":1961},[635],[573,1963,1532],{"className":1964},[629,639],[573,1966,1535],{"className":1967},[1586],[573,1969],{"className":1970,"style":1323},[1004],[573,1972,1538],{"className":1973},[629,639],[573,1975,1535],{"className":1976},[1586],[573,1978],{"className":1979,"style":1323},[1004],[573,1981,1543],{"className":1982},[629,639],[573,1984,1546],{"className":1985},[644],[564,1987,1988,1989,1135],{},"Nach Größe sortiert sind das ",[573,1990,1992,2024],{"className":1991},[576],[573,1993,1995],{"className":1994},[580],[582,1996,1997],{"xmlns":584},[586,1998,1999,2021],{},[589,2000,2001,2004,2007,2009,2011,2013,2015,2017,2019],{},[857,2002,2003],{},"1",[597,2005,2006],{},"+",[857,2008,1624],{},[597,2010,2006],{},[857,2012,1624],{},[597,2014,2006],{},[857,2016,2003],{},[597,2018,978],{},[857,2020,1629],{},[608,2022,2023],{"encoding":610},"1 + 3 + 3 + 1 = 8",[573,2025,2027,2046,2064,2082,2100],{"className":2026,"ariaHidden":616},[615],[573,2028,2030,2034,2037,2040,2043],{"className":2029},[620],[573,2031],{"className":2032,"style":2033},[624],"height:0.7278em;vertical-align:-0.0833em;",[573,2035,2003],{"className":2036},[629],[573,2038],{"className":2039,"style":1005},[1004],[573,2041,2006],{"className":2042},[1009],[573,2044],{"className":2045,"style":1005},[1004],[573,2047,2049,2052,2055,2058,2061],{"className":2048},[620],[573,2050],{"className":2051,"style":2033},[624],[573,2053,1624],{"className":2054},[629],[573,2056],{"className":2057,"style":1005},[1004],[573,2059,2006],{"className":2060},[1009],[573,2062],{"className":2063,"style":1005},[1004],[573,2065,2067,2070,2073,2076,2079],{"className":2066},[620],[573,2068],{"className":2069,"style":2033},[624],[573,2071,1624],{"className":2072},[629],[573,2074],{"className":2075,"style":1005},[1004],[573,2077,2006],{"className":2078},[1009],[573,2080],{"className":2081,"style":1005},[1004],[573,2083,2085,2088,2091,2094,2097],{"className":2084},[620],[573,2086],{"className":2087,"style":997},[624],[573,2089,2003],{"className":2090},[629],[573,2092],{"className":2093,"style":1063},[1004],[573,2095,978],{"className":2096},[1067],[573,2098],{"className":2099,"style":1063},[1004],[573,2101,2103,2106],{"className":2102},[620],[573,2104],{"className":2105,"style":997},[624],[573,2107,1629],{"className":2108},[629],[564,2110,2111,2112,2141,2142,2227],{},"Eine Pizzeria bietet ",[573,2113,2115,2129],{"className":2114},[576],[573,2116,2118],{"className":2117},[580],[582,2119,2120],{"xmlns":584},[586,2121,2122,2127],{},[589,2123,2124],{},[857,2125,2126],{},"10",[608,2128,2126],{"encoding":610},[573,2130,2132],{"className":2131,"ariaHidden":616},[615],[573,2133,2135,2138],{"className":2134},[620],[573,2136],{"className":2137,"style":997},[624],[573,2139,2126],{"className":2140},[629]," Beläge an. Jede Auswahl von Belägen, auch gar keiner, ergibt eine andere Pizza: ",[573,2143,2145,2168],{"className":2144},[576],[573,2146,2148],{"className":2147},[580],[582,2149,2150],{"xmlns":584},[586,2151,2152,2165],{},[589,2153,2154,2160,2162],{},[854,2155,2156,2158],{},[857,2157,859],{},[857,2159,2126],{},[597,2161,978],{},[857,2163,2164],{},"1024",[608,2166,2167],{"encoding":610},"2^{10} = 1024",[573,2169,2171,2218],{"className":2170,"ariaHidden":616},[615],[573,2172,2174,2177,2209,2212,2215],{"className":2173},[620],[573,2175],{"className":2176,"style":1642},[624],[573,2178,2180,2183],{"className":2179},[629],[573,2181,859],{"className":2182},[629],[573,2184,2186],{"className":2185},[884],[573,2187,2189],{"className":2188},[888],[573,2190,2192],{"className":2191},[892],[573,2193,2195],{"className":2194,"style":1642},[896],[573,2196,2197,2200],{"style":899},[573,2198],{"className":2199,"style":904},[903],[573,2201,2203],{"className":2202},[908,909,910,911],[573,2204,2206],{"className":2205},[629,911],[573,2207,2126],{"className":2208},[629,911],[573,2210],{"className":2211,"style":1063},[1004],[573,2213,978],{"className":2214},[1067],[573,2216],{"className":2217,"style":1063},[1004],[573,2219,2221,2224],{"className":2220},[620],[573,2222],{"className":2223,"style":997},[624],[573,2225,2164],{"className":2226},[629]," Möglichkeiten.",[564,2229,2230,2231,1605,2294,915],{},"Die Potenzmenge von ",[573,2232,2234,2262],{"className":2233},[576],[573,2235,2237],{"className":2236},[580],[582,2238,2239],{"xmlns":584},[586,2240,2241,2259],{},[589,2242,2243,2245,2247,2249,2251,2253,2255,2257],{},[592,2244,595],{"mathvariant":594},[597,2246,600],{"stretchy":599},[597,2248,1529],{"stretchy":599},[857,2250,2003],{},[597,2252,1535],{"separator":616},[857,2254,859],{},[597,2256,1546],{"stretchy":599},[597,2258,606],{"stretchy":599},[608,2260,2261],{"encoding":610},"\\mathcal{P}(\\{1, 2\\})",[573,2263,2265],{"className":2264,"ariaHidden":616},[615],[573,2266,2268,2271,2274,2278,2281,2284,2287,2290],{"className":2267},[620],[573,2269],{"className":2270,"style":625},[624],[573,2272,595],{"className":2273,"style":631},[629,630],[573,2275,2277],{"className":2276},[635],"({",[573,2279,2003],{"className":2280},[629],[573,2282,1535],{"className":2283},[1586],[573,2285],{"className":2286,"style":1323},[1004],[573,2288,859],{"className":2289},[629],[573,2291,2293],{"className":2292},[644],"})",[573,2295,2297,2324],{"className":2296},[576],[573,2298,2300],{"className":2299},[580],[582,2301,2302],{"xmlns":584},[586,2303,2304,2321],{},[589,2305,2306,2316,2318],{},[854,2307,2308,2310],{},[857,2309,859],{},[854,2311,2312,2314],{},[857,2313,859],{},[857,2315,859],{},[597,2317,978],{},[857,2319,2320],{},"16",[608,2322,2323],{"encoding":610},"2^{2^2} = 16",[573,2325,2327,2406],{"className":2326,"ariaHidden":616},[615],[573,2328,2330,2334,2397,2400,2403],{"className":2329},[620],[573,2331],{"className":2332,"style":2333},[624],"height:0.9869em;",[573,2335,2337,2340],{"className":2336},[629],[573,2338,859],{"className":2339},[629],[573,2341,2343],{"className":2342},[884],[573,2344,2346],{"className":2345},[888],[573,2347,2349],{"className":2348},[892],[573,2350,2352],{"className":2351,"style":2333},[896],[573,2353,2354,2357],{"style":899},[573,2355],{"className":2356,"style":904},[903],[573,2358,2360],{"className":2359},[908,909,910,911],[573,2361,2363],{"className":2362},[629,911],[573,2364,2366,2369],{"className":2365},[629,911],[573,2367,859],{"className":2368},[629,911],[573,2370,2372],{"className":2371},[884],[573,2373,2375],{"className":2374},[888],[573,2376,2378],{"className":2377},[892],[573,2379,2382],{"className":2380,"style":2381},[896],"height:0.8913em;",[573,2383,2385,2389],{"style":2384},"top:-2.931em;margin-right:0.0714em;",[573,2386],{"className":2387,"style":2388},[903],"height:2.5em;",[573,2390,2394],{"className":2391},[908,2392,2393,911],"reset-size3","size1",[573,2395,859],{"className":2396},[629,911],[573,2398],{"className":2399,"style":1063},[1004],[573,2401,978],{"className":2402},[1067],[573,2404],{"className":2405,"style":1063},[1004],[573,2407,2409,2412],{"className":2408},[620],[573,2410],{"className":2411,"style":997},[624],[573,2413,2320],{"className":2414},[629],[559,2416,2418],{"id":2417},"typische-fehler","Typische Fehler",[2420,2421,2422,2670,2879],"ul",{},[2423,2424,2425,2457,2458,2461,2462,2545,2546,2611,2612,2641,2642,1135],"li",{},[568,2426,2427,2428,2456],{},"Leere Menge oder ",[573,2429,2431,2444],{"className":2430},[576],[573,2432,2434],{"className":2433},[580],[582,2435,2436],{"xmlns":584},[586,2437,2438,2442],{},[589,2439,2440],{},[592,2441,603],{},[608,2443,603],{"encoding":610},[573,2445,2447],{"className":2446,"ariaHidden":616},[615],[573,2448,2450,2453],{"className":2449},[620],[573,2451],{"className":2452,"style":671},[624],[573,2454,603],{"className":2455,"style":640},[629,639]," selbst vergessen."," Wer nur die ",[568,2459,2460],{},"echten, nichtleeren"," Teilmengen zählt, erhält ",[573,2463,2465,2488],{"className":2464},[576],[573,2466,2468],{"className":2467},[580],[582,2469,2470],{"xmlns":584},[586,2471,2472,2485],{},[589,2473,2474,2480,2483],{},[854,2475,2476,2478],{},[857,2477,859],{},[592,2479,781],{},[597,2481,2482],{},"−",[857,2484,859],{},[608,2486,2487],{"encoding":610},"2^n - 2",[573,2489,2491,2536],{"className":2490,"ariaHidden":616},[615],[573,2492,2494,2498,2527,2530,2533],{"className":2493},[620],[573,2495],{"className":2496,"style":2497},[624],"height:0.7477em;vertical-align:-0.0833em;",[573,2499,2501,2504],{"className":2500},[629],[573,2502,859],{"className":2503},[629],[573,2505,2507],{"className":2506},[884],[573,2508,2510],{"className":2509},[888],[573,2511,2513],{"className":2512},[892],[573,2514,2516],{"className":2515,"style":874},[896],[573,2517,2518,2521],{"style":899},[573,2519],{"className":2520,"style":904},[903],[573,2522,2524],{"className":2523},[908,909,910,911],[573,2525,781],{"className":2526},[629,639,911],[573,2528],{"className":2529,"style":1005},[1004],[573,2531,2482],{"className":2532},[1009],[573,2534],{"className":2535,"style":1005},[1004],[573,2537,2539,2542],{"className":2538},[620],[573,2540],{"className":2541,"style":997},[624],[573,2543,859],{"className":2544},[629],", bei ",[573,2547,2549,2575],{"className":2548},[576],[573,2550,2552],{"className":2551},[580],[582,2553,2554],{"xmlns":584},[586,2555,2556,2572],{},[589,2557,2558,2560,2562,2564,2566,2568,2570],{},[597,2559,1529],{"stretchy":599},[592,2561,1532],{},[597,2563,1535],{"separator":616},[592,2565,1538],{},[597,2567,1535],{"separator":616},[592,2569,1543],{},[597,2571,1546],{"stretchy":599},[608,2573,2574],{"encoding":610},"\\{a, b, c\\}",[573,2576,2578],{"className":2577,"ariaHidden":616},[615],[573,2579,2581,2584,2587,2590,2593,2596,2599,2602,2605,2608],{"className":2580},[620],[573,2582],{"className":2583,"style":625},[624],[573,2585,1529],{"className":2586},[635],[573,2588,1532],{"className":2589},[629,639],[573,2591,1535],{"className":2592},[1586],[573,2594],{"className":2595,"style":1323},[1004],[573,2597,1538],{"className":2598},[629,639],[573,2600,1535],{"className":2601},[1586],[573,2603],{"className":2604,"style":1323},[1004],[573,2606,1543],{"className":2607},[629,639],[573,2609,1546],{"className":2610},[644]," also ",[573,2613,2615,2629],{"className":2614},[576],[573,2616,2618],{"className":2617},[580],[582,2619,2620],{"xmlns":584},[586,2621,2622,2627],{},[589,2623,2624],{},[857,2625,2626],{},"6",[608,2628,2626],{"encoding":610},[573,2630,2632],{"className":2631,"ariaHidden":616},[615],[573,2633,2635,2638],{"className":2634},[620],[573,2636],{"className":2637,"style":997},[624],[573,2639,2626],{"className":2640},[629]," statt ",[573,2643,2645,2658],{"className":2644},[576],[573,2646,2648],{"className":2647},[580],[582,2649,2650],{"xmlns":584},[586,2651,2652,2656],{},[589,2653,2654],{},[857,2655,1629],{},[608,2657,1629],{"encoding":610},[573,2659,2661],{"className":2660,"ariaHidden":616},[615],[573,2662,2664,2667],{"className":2663},[620],[573,2665],{"className":2666,"style":997},[624],[573,2668,1629],{"className":2669},[629],[2423,2671,2672,2719,2720,2796,2797,2878],{},[568,2673,2674,2718],{},[573,2675,2677,2697],{"className":2676},[576],[573,2678,2680],{"className":2679},[580],[582,2681,2682],{"xmlns":584},[586,2683,2684,2694],{},[589,2685,2686,2688,2690,2692],{},[592,2687,595],{"mathvariant":594},[597,2689,600],{"stretchy":599},[592,2691,691],{"mathvariant":690},[597,2693,606],{"stretchy":599},[608,2695,2696],{"encoding":610},"\\mathcal{P}(\\emptyset)",[573,2698,2700],{"className":2699,"ariaHidden":616},[615],[573,2701,2703,2706,2709,2712,2715],{"className":2702},[620],[573,2704],{"className":2705,"style":625},[624],[573,2707,595],{"className":2708,"style":631},[629,630],[573,2710,600],{"className":2711},[635],[573,2713,691],{"className":2714},[629],[573,2716,606],{"className":2717},[644]," für leer halten."," Die leere Menge hat genau eine Teilmenge, sich selbst: ",[573,2721,2723,2751],{"className":2722},[576],[573,2724,2726],{"className":2725},[580],[582,2727,2728],{"xmlns":584},[586,2729,2730,2748],{},[589,2731,2732,2734,2736,2738,2740,2742,2744,2746],{},[592,2733,595],{"mathvariant":594},[597,2735,600],{"stretchy":599},[592,2737,691],{"mathvariant":690},[597,2739,606],{"stretchy":599},[597,2741,978],{},[597,2743,1529],{"stretchy":599},[592,2745,691],{"mathvariant":690},[597,2747,1546],{"stretchy":599},[608,2749,2750],{"encoding":610},"\\mathcal{P}(\\emptyset) = \\{\\emptyset\\}",[573,2752,2754,2781],{"className":2753,"ariaHidden":616},[615],[573,2755,2757,2760,2763,2766,2769,2772,2775,2778],{"className":2756},[620],[573,2758],{"className":2759,"style":625},[624],[573,2761,595],{"className":2762,"style":631},[629,630],[573,2764,600],{"className":2765},[635],[573,2767,691],{"className":2768},[629],[573,2770,606],{"className":2771},[644],[573,2773],{"className":2774,"style":1063},[1004],[573,2776,978],{"className":2777},[1067],[573,2779],{"className":2780,"style":1063},[1004],[573,2782,2784,2787,2790,2793],{"className":2783},[620],[573,2785],{"className":2786,"style":625},[624],[573,2788,1529],{"className":2789},[635],[573,2791,691],{"className":2792},[629],[573,2794,1546],{"className":2795},[644]," mit ",[573,2798,2800,2822],{"className":2799},[576],[573,2801,2803],{"className":2802},[580],[582,2804,2805],{"xmlns":584},[586,2806,2807,2819],{},[589,2808,2809,2815,2817],{},[854,2810,2811,2813],{},[857,2812,859],{},[857,2814,1202],{},[597,2816,978],{},[857,2818,2003],{},[608,2820,2821],{"encoding":610},"2^0 = 1",[573,2823,2825,2869],{"className":2824,"ariaHidden":616},[615],[573,2826,2828,2831,2860,2863,2866],{"className":2827},[620],[573,2829],{"className":2830,"style":1642},[624],[573,2832,2834,2837],{"className":2833},[629],[573,2835,859],{"className":2836},[629],[573,2838,2840],{"className":2839},[884],[573,2841,2843],{"className":2842},[888],[573,2844,2846],{"className":2845},[892],[573,2847,2849],{"className":2848,"style":1642},[896],[573,2850,2851,2854],{"style":899},[573,2852],{"className":2853,"style":904},[903],[573,2855,2857],{"className":2856},[908,909,910,911],[573,2858,1202],{"className":2859},[629,911],[573,2861],{"className":2862,"style":1063},[1004],[573,2864,978],{"className":2865},[1067],[573,2867],{"className":2868,"style":1063},[1004],[573,2870,2872,2875],{"className":2871},[620],[573,2873],{"className":2874,"style":997},[624],[573,2876,2003],{"className":2877},[629]," Element.",[2423,2880,2881,571,2884,2937,2938,708,3014,1135],{},[568,2882,2883],{},"Elemente und Teilmengen verwechseln.",[573,2885,2887,2906],{"className":2886},[576],[573,2888,2890],{"className":2889},[580],[582,2891,2892],{"xmlns":584},[586,2893,2894,2903],{},[589,2895,2896,2898,2901],{},[592,2897,1532],{},[597,2899,2900],{},"∈",[592,2902,603],{},[608,2904,2905],{"encoding":610},"a \\in M",[573,2907,2909,2928],{"className":2908,"ariaHidden":616},[615],[573,2910,2912,2916,2919,2922,2925],{"className":2911},[620],[573,2913],{"className":2914,"style":2915},[624],"height:0.5782em;vertical-align:-0.0391em;",[573,2917,1532],{"className":2918},[629,639],[573,2920],{"className":2921,"style":1063},[1004],[573,2923,2900],{"className":2924},[1067],[573,2926],{"className":2927,"style":1063},[1004],[573,2929,2931,2934],{"className":2930},[620],[573,2932],{"className":2933,"style":671},[624],[573,2935,603],{"className":2936,"style":640},[629,639],", aber ",[573,2939,2941,2969],{"className":2940},[576],[573,2942,2944],{"className":2943},[580],[582,2945,2946],{"xmlns":584},[586,2947,2948,2966],{},[589,2949,2950,2952,2954,2956,2958,2960,2962,2964],{},[597,2951,1529],{"stretchy":599},[592,2953,1532],{},[597,2955,1546],{"stretchy":599},[597,2957,2900],{},[592,2959,595],{"mathvariant":594},[597,2961,600],{"stretchy":599},[592,2963,603],{},[597,2965,606],{"stretchy":599},[608,2967,2968],{"encoding":610},"\\{a\\} \\in \\mathcal{P}(M)",[573,2970,2972,2996],{"className":2971,"ariaHidden":616},[615],[573,2973,2975,2978,2981,2984,2987,2990,2993],{"className":2974},[620],[573,2976],{"className":2977,"style":625},[624],[573,2979,1529],{"className":2980},[635],[573,2982,1532],{"className":2983},[629,639],[573,2985,1546],{"className":2986},[644],[573,2988],{"className":2989,"style":1063},[1004],[573,2991,2900],{"className":2992},[1067],[573,2994],{"className":2995,"style":1063},[1004],[573,2997,2999,3002,3005,3008,3011],{"className":2998},[620],[573,3000],{"className":3001,"style":625},[624],[573,3003,595],{"className":3004,"style":631},[629,630],[573,3006,600],{"className":3007},[635],[573,3009,603],{"className":3010,"style":640},[629,639],[573,3012,606],{"className":3013},[644],[573,3015,3017,3040],{"className":3016},[576],[573,3018,3020],{"className":3019},[580],[582,3021,3022],{"xmlns":584},[586,3023,3024,3037],{},[589,3025,3026,3028,3030,3032,3035],{},[597,3027,1529],{"stretchy":599},[592,3029,1532],{},[597,3031,1546],{"stretchy":599},[597,3033,3034],{},"⊆",[592,3036,603],{},[608,3038,3039],{"encoding":610},"\\{a\\} \\subseteq M",[573,3041,3043,3067],{"className":3042,"ariaHidden":616},[615],[573,3044,3046,3049,3052,3055,3058,3061,3064],{"className":3045},[620],[573,3047],{"className":3048,"style":625},[624],[573,3050,1529],{"className":3051},[635],[573,3053,1532],{"className":3054},[629,639],[573,3056,1546],{"className":3057},[644],[573,3059],{"className":3060,"style":1063},[1004],[573,3062,3034],{"className":3063},[1067],[573,3065],{"className":3066,"style":1063},[1004],[573,3068,3070,3073],{"className":3069},[620],[573,3071],{"className":3072,"style":671},[624],[573,3074,603],{"className":3075,"style":640},[629,639],{"title":3077,"searchDepth":3078,"depth":3078,"links":3079},"",2,[3080,3081,3082],{"id":561,"depth":3078,"text":562},{"id":1507,"depth":3078,"text":1508},{"id":2417,"depth":3078,"text":2418},[3084,3087,3089,3091,3093,3095,3097],{"lhs":3085,"rhs":3086},"combinations(3, 0) + combinations(3, 1) + combinations(3, 2) + combinations(3, 3)","2^3",{"lhs":3088,"rhs":1629},"1 + 3 + 3 + 1",{"lhs":3090,"rhs":2164},"sum(map(0:10, f(k) = combinations(10, k)))",{"lhs":3092,"rhs":2164},"2^10",{"lhs":3094,"rhs":2003},"2^0",{"lhs":3096,"rhs":2320},"2^(2^2)",{"lhs":3098,"rhs":2626},"2^3 - 2","M \\text{ endliche Menge}","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>M\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">M\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">M\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">endliche Menge\u003C\u002Fspan>",null,"md","power-set-cardinality","|\\mathcal{P}(M)| = 2^{|M|}","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003Cmi mathvariant=\"script\">P\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>M\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmsup>\u003Cmn>2\u003C\u002Fmn>\u003Cmrow>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003Cmi>M\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">|\\mathcal{P}(M)| = 2^{|M|}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">∣\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0822em;\">P\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">M\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mord\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.938em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">2\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.938em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">∣\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.109em;\">M\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">∣\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fmaechtigkeit-der-potenzmenge",[],[3112,3113,3114],"binomial-coefficient","binomial-identities","tuples-with-repetition",{"title":548,"description":3077},"de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fmaechtigkeit-der-potenzmenge","Eine Menge mit n Elementen hat genau 2ⁿ Teilmengen, die leere Menge und die Menge selbst eingeschlossen.",[3119,3120,3121,3122,3123],"mengenlehre","potenzmenge","teilmengen","mächtigkeit","kombinatorik",[3125,3128,3131],{"symbol":603,"meaning":3126,"symbolHtml":3127},"endliche Menge","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>M\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">M\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">M\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":611,"meaning":3129,"symbolHtml":3130},"Potenzmenge von M, die Menge aller Teilmengen von M","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"script\">P\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>M\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathcal{P}(M)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0822em;\">P\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">M\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":3132,"meaning":3133,"symbolHtml":3134},"|M|","Mächtigkeit, also Anzahl der Elemente von M","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003Cmi>M\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">|M|\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">∣\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">M\u003C\u002Fspan>\u003Cspan class=\"mord\">∣\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","AnmXHpC7xmEYS-Vy2STHp0q70ack2AcAPcl-l3P7kJM",[3137,3140,3143],{"stem":3138,"title":3139,"category":187},"de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Frechenregeln-binomialkoeffizienten","Rechenregeln für Binomialkoeffizienten",{"stem":3141,"title":3142,"category":441},"de\u002Fformulas\u002Fmath\u002Fstochastics\u002Fbinomialkoeffizient","Binomialkoeffizient",{"stem":3144,"title":3145,"category":441},"de\u002Fformulas\u002Fmath\u002Fstochastics\u002Fvariation-mit-wiederholung","Variation mit Wiederholung",1793824276633]