4806:
4022:
3984:
3977:
3892:
3885:
3878:
3803:
3796:
3789:
3776:
3769:
3762:
3708:
3701:
3694:
3681:
3674:
3596:
3553:
3546:
3525:
3518:
3511:
3379:
3372:
3308:
3287:
3233:
3226:
3212:
3158:
3151:
3144:
3137:
3090:
3083:
3076:
3069:
3062:
3055:
3048:
2953:
2910:
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2875:
2868:
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2790:
2783:
2776:
2769:
2762:
2736:
2729:
2722:
2715:
2708:
2701:
2656:
2649:
2642:
2635:
2628:
2621:
2591:
2584:
2577:
2570:
2563:
2556:
2549:
2542:
3991:
3899:
4740:
4036:
4029:
3970:
3963:
3956:
3937:
3930:
3871:
3864:
3857:
3841:
3746:
3667:
3648:
3610:
3603:
3583:
3576:
3569:
3539:
3532:
3504:
3497:
3490:
3471:
3464:
3457:
3450:
3443:
3436:
3429:
3422:
3415:
3393:
3386:
3365:
3358:
3351:
3344:
3337:
3315:
3301:
3294:
3280:
3273:
3266:
3259:
3240:
3219:
3205:
3198:
3191:
3184:
3165:
3130:
3123:
3116:
3109:
3041:
3034:
3027:
3005:
2998:
2946:
2896:
2743:
2694:
2687:
2663:
2614:
2607:
2535:
2528:
1345:, specifying which points belong to which lines. An incidence structure can be specified by a family of sets (even if two distinct lines contain the same set of points), the sets of points belonging to each line, and any family of sets can be interpreted as an incidence structure in this way.
1322:, joined face to face. In an abstract simplicial complex, each simplex is represented simply as the set of its vertices. Any family of finite sets without repetitions in which the subsets of any set in the family also belong to the family forms an abstract simplicial complex.
1140:
879:
1307:, each of which may be an arbitrary set. The hyperedges of a hypergraph form a family of sets, and any family of sets can be interpreted as a hypergraph that has the union of the sets as its vertices.
763:
3835:
3740:
3642:
4510:
1199:
2502:
2436:
2387:
2469:
4453:
4073:
2160:
1289:
Certain types of objects from other areas of mathematics are equivalent to families of sets, in that they can be described purely as a collection of sets of objects of some type:
1280:
955:
1250:
4364:
2338:
933:
654:
4395:
4329:
1871:
1167:
4477:
4102:
2986:
2855:
2823:
2189:
2079:
1844:
1820:
1790:
1223:
1083:
4298:
4239:
2940:
2309:
2221:
1521:
1401:
5263:
354:
4213:
4169:
2280:
2251:
2101:
1027:
572:
386:
4014:
3922:
1445:
1992:
1705:
1474:
517:
429:
158:
1969:
1945:
1727:
1678:
1646:
1626:
1606:
1586:
1541:
1498:
1421:
995:
674:
592:
538:
492:
471:
451:
406:
322:
298:
260:
230:
210:
190:
131:
107:
80:
1088:
2121:
5413:
4723:
1364:. A block code can also be described as a family of sets, by describing each codeword as the set of positions at which it contains a 1.
4952:
4772:
1761:
232:. In some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a
5280:
4705:
4687:
768:
5258:
683:
2114:
5032:
4911:
5275:
3810:
3715:
3617:
4246:
2508:
5268:
1680:
lies in only finitely many members of the family. If every point of a cover lies in exactly one member, the cover is a
4906:
4869:
4482:
1923:
1794:
1750:
1311:
1055:
1172:
2477:
1047:
1054:, gives necessary and sufficient conditions for a finite family of non-empty sets (repetitions allowed) to have a
5418:
4957:
4849:
4837:
4832:
2395:
2346:
2107:
1745:
1353:
4765:
2444:
1553:
4744:
4400:
5377:
5295:
5170:
5122:
4936:
4859:
1656:
4051:
2138:
2002:
5329:
5210:
5022:
4842:
4558:
1255:
940:
1228:
5245:
5159:
5079:
5059:
5037:
4343:
4104:
2317:
2191:
1361:
1298:
884:
600:
5319:
5309:
5143:
5074:
5027:
4967:
4854:
4628:, p.34. The University Series in Undergraduate Mathematics, 1960. Litton Educational Publishing, Inc.
4588:
4540:
4373:
4307:
3947:
3848:
1894:
1849:
1449:
1145:
4458:
4083:
2961:
2830:
2798:
2170:
2060:
1825:
1801:
1771:
1204:
1064:
5314:
5225:
5138:
5133:
5128:
4942:
4884:
4822:
4758:
3560:
2034:
2010:
1755:
1557:
1326:
269:
4277:
4224:
2919:
2288:
2205:
1506:
5237:
5232:
5017:
4972:
4879:
4534:
2046:
2027:
1681:
1315:
1037:
966:
233:
59:
4537: – Collection of sets in mathematics that can be defined based on a property of its members
1907:
272:
concerns the largest and smallest examples of families of sets satisfying certain restrictions.
1374:
5094:
4931:
4923:
4894:
4864:
4795:
4719:
4701:
4683:
4112:
1730:
1368:
327:
87:
47:
4198:
4154:
2259:
2230:
2086:
1003:
544:
362:
5382:
5372:
5357:
5352:
5220:
4874:
1564:
1357:
3999:
3907:
1430:
1318:, a shape formed by unions of line segments, triangles, tetrahedra, and higher-dimensional
5251:
5189:
5007:
4827:
4528:
2018:
1948:
1915:
1906:
is a set family such that any minimal subfamily with empty intersection has bounded size.
1875:
1338:
1041:
1033:
1135:{\displaystyle \cup {\mathcal {F}}:={\textstyle \bigcup \limits _{F\in {\mathcal {F}}}}F}
1974:
1687:
1456:
499:
411:
140:
5387:
5184:
5165:
5069:
5011:
4947:
4889:
4564:
4367:
4301:
3753:
2753:
1954:
1930:
1889:
1712:
1663:
1631:
1611:
1591:
1571:
1526:
1483:
1406:
980:
958:
659:
577:
523:
477:
456:
436:
391:
307:
283:
245:
215:
195:
175:
116:
92:
65:
5407:
5392:
5362:
5194:
5108:
5103:
4570:
4552:
3325:
2991:
2882:
5342:
5337:
5155:
5084:
5042:
4901:
4805:
4582:
4555: – Algebraic concept in measure theory, also referred to as an algebra of sets
4335:
4262:
3097:
3015:
2674:
2599:
2199:
1902:
4598:
4567: – Collection of objects, each associated with an element from some index set
3403:
5367:
5002:
2042:
2023:
1051:
51:
35:
5347:
5215:
5118:
4781:
4604:
4546:
3481:
3250:
3175:
1911:
1739:
1543:
itself, and is closed under arbitrary set unions and finite set intersections.
1349:
1303:
1294:
520:
474:
264:
240:
31:
5150:
5113:
5064:
4962:
4576:
4268:
3655:
2517:
1501:
998:
357:
301:
4739:
17:
2038:
2000:
is an abstract simplicial complex with an additional property called the
1765:
is a family that is the union of countably many locally finite families.
1734:
1650:
1356:
of 0s and 1s, all the same length. When each pair of codewords has large
677:
172:. Additionally, a family of sets may be defined as a function from a set
55:
3990:
3898:
1996:
1319:
1893:
is a set family in which none of the sets contains any of the others.
5175:
4997:
4573: – Family closed under complements and countable disjoint unions
83:
2022:
is a set family closed under arbitrary intersections and unions of
5047:
4814:
1608:
belongs to some member of the family. A subfamily of a cover of
212:, in which case the sets of the family are indexed by members of
160:
More generally, a collection of any sets whatsoever is called a
4754:
965:
family of sets. That is, it is not itself a set but instead a
46:) can mean, depending upon the context, any of the following:
4750:
4561: – Expression denoting a set of sets in formal semantics
4488:
4464:
4379:
4313:
4089:
4057:
3822:
3727:
3629:
2489:
2456:
2176:
2144:
2066:
1855:
1831:
1807:
1777:
1753:
that intersects only finitely many members of the family. A
1264:
1237:
1210:
1151:
1120:
1097:
1070:
874:{\displaystyle A_{1}=\{a,b,c\},A_{2}=\{1,2\},A_{3}=\{1,2\},}
4585: – Family closed under unions and relative complements
758:{\displaystyle F=\left\{A_{1},A_{2},A_{3},A_{4}\right\},}
4700:(5th ed.), Upper Saddle River, NJ: Prentice Hall,
1106:
4485:
4461:
4403:
4376:
4346:
4310:
4280:
4227:
4201:
4157:
4086:
4054:
4002:
3910:
3813:
3718:
3620:
2964:
2922:
2833:
2801:
2480:
2447:
2398:
2349:
2320:
2291:
2262:
2233:
2208:
2173:
2141:
2089:
2063:
1977:
1957:
1933:
1852:
1828:
1804:
1774:
1715:
1690:
1666:
1634:
1614:
1594:
1574:
1529:
1509:
1486:
1476:
which is a family of sets (whose elements are called
1459:
1433:
1409:
1377:
1258:
1231:
1207:
1175:
1148:
1091:
1067:
1006:
983:
943:
887:
771:
686:
662:
603:
580:
547:
526:
502:
480:
459:
439:
414:
394:
365:
330:
310:
286:
248:
218:
198:
178:
143:
119:
95:
68:
4607: – Family of sets closed under countable unions
5328:
5291:
5203:
5093:
4981:
4922:
4813:
4788:
4531: – Identities and relationships involving sets
3830:{\displaystyle \varnothing \not \in {\mathcal {F}}}
3735:{\displaystyle \varnothing \not \in {\mathcal {F}}}
3637:{\displaystyle \varnothing \not \in {\mathcal {F}}}
1352:consists of a set of codewords, each of which is a
4504:
4471:
4447:
4389:
4358:
4323:
4292:
4233:
4207:
4163:
4096:
4067:
4008:
3916:
3829:
3734:
3636:
2980:
2934:
2849:
2817:
2496:
2463:
2430:
2381:
2332:
2303:
2274:
2245:
2215:
2183:
2154:
2095:
2073:
1986:
1963:
1939:
1865:
1838:
1814:
1784:
1721:
1699:
1672:
1640:
1620:
1600:
1580:
1535:
1515:
1492:
1468:
1439:
1415:
1395:
1314:is a combinatorial abstraction of the notion of a
1297:, also called a set system, is formed by a set of
1274:
1244:
1217:
1193:
1161:
1134:
1077:
1021:
989:
949:
927:
873:
757:
668:
648:
586:
566:
532:
511:
486:
465:
445:
423:
400:
380:
348:
316:
292:
254:
224:
204:
184:
152:
125:
101:
74:
4549: – Ring closed under countable intersections
4579: – Family of sets closed under intersection
4505:{\displaystyle {\mathcal {F}}\neq \varnothing .}
1194:{\displaystyle \cup \varnothing =\varnothing .}
2497:{\displaystyle \varnothing \in {\mathcal {F}}}
1554:List of partition topics § Set partitions
4766:
4649:
2115:
1897:bounds the maximum size of a Sperner family.
8:
4543: – Symmetric arrangement of finite sets
1032:Any family of sets without repetitions is a
919:
901:
865:
853:
834:
822:
803:
785:
640:
610:
2431:{\displaystyle A_{1}\cup A_{2}\cup \cdots }
2382:{\displaystyle A_{1}\cap A_{2}\cap \cdots }
27:Any collection of sets, or subsets of a set
4773:
4759:
4751:
4601: – Algebraic structure of set algebra
4593:Set of sets that do not contain themselves
2122:
2108:
2051:
1918:of bounded dimension form Helly families.
4487:
4486:
4484:
4463:
4462:
4460:
4433:
4420:
4402:
4378:
4377:
4375:
4345:
4312:
4311:
4309:
4279:
4226:
4200:
4156:
4088:
4087:
4085:
4056:
4055:
4053:
4001:
3909:
3821:
3820:
3812:
3726:
3725:
3717:
3628:
3627:
3619:
2969:
2963:
2921:
2838:
2832:
2806:
2800:
2488:
2487:
2479:
2464:{\displaystyle \Omega \in {\mathcal {F}}}
2455:
2454:
2446:
2416:
2403:
2397:
2367:
2354:
2348:
2319:
2290:
2261:
2232:
2209:
2207:
2175:
2174:
2172:
2143:
2142:
2140:
2088:
2065:
2064:
2062:
1976:
1956:
1932:
1854:
1853:
1851:
1830:
1829:
1827:
1806:
1805:
1803:
1776:
1775:
1773:
1714:
1689:
1665:
1633:
1613:
1593:
1573:
1528:
1508:
1485:
1458:
1432:
1408:
1376:
1263:
1262:
1257:
1236:
1235:
1230:
1209:
1208:
1206:
1174:
1150:
1149:
1147:
1119:
1118:
1111:
1105:
1096:
1095:
1090:
1069:
1068:
1066:
1005:
982:
942:
892:
886:
844:
813:
776:
770:
741:
728:
715:
702:
685:
661:
602:
579:
552:
546:
525:
501:
479:
458:
438:
413:
393:
364:
329:
309:
285:
247:
217:
197:
177:
142:
118:
94:
67:
4714:Roberts, Fred S.; Tesman, Barry (2009),
4718:(2nd ed.), Boca Raton: CRC Press,
4637:
4617:
4496:
4448:{\displaystyle A,B,A_{1},A_{2},\ldots }
4350:
4284:
4228:
3814:
3719:
3621:
2481:
2324:
2295:
2105:
1510:
1252:and also a family over any superset of
1185:
1179:
280:The set of all subsets of a given set
4661:
4340:is a semiring where every complement
4068:{\displaystyle {\mathcal {F}}\colon }
2155:{\displaystyle {\mathcal {F}}\colon }
7:
1951:; that is, every subset of a set in
1947:(consisting of finite sets) that is
1879:is a particular type of refinement.
1423:is a set (whose elements are called
1275:{\displaystyle \cup {\mathcal {F}}.}
950:{\displaystyle \operatorname {Ord} }
656:An example of a family of sets over
2033:Other examples of set families are
1762:countably locally finite collection
1245:{\displaystyle \cup {\mathcal {F}}}
1108:
4591: – Paradox in set theory (or
4359:{\displaystyle \Omega \setminus A}
4347:
4202:
4158:
2448:
2333:{\displaystyle \Omega \setminus A}
2321:
2090:
1056:system of distinct representatives
1007:
366:
331:
25:
1749:if each point in the space has a
1142:denotes the union of all sets in
4804:
4738:
4034:
4027:
4020:
3989:
3982:
3975:
3968:
3961:
3954:
3935:
3928:
3897:
3890:
3883:
3876:
3869:
3862:
3855:
3839:
3801:
3794:
3787:
3774:
3767:
3760:
3744:
3706:
3699:
3692:
3679:
3672:
3665:
3646:
3608:
3601:
3594:
3581:
3574:
3567:
3551:
3544:
3537:
3530:
3523:
3516:
3509:
3502:
3495:
3488:
3469:
3462:
3455:
3448:
3441:
3434:
3427:
3420:
3413:
3391:
3384:
3377:
3370:
3363:
3356:
3349:
3342:
3335:
3313:
3306:
3299:
3292:
3285:
3278:
3271:
3264:
3257:
3238:
3231:
3224:
3217:
3210:
3203:
3196:
3189:
3182:
3163:
3156:
3149:
3142:
3135:
3128:
3121:
3114:
3107:
3088:
3081:
3074:
3067:
3060:
3053:
3046:
3039:
3032:
3025:
3003:
2996:
2951:
2944:
2908:
2901:
2894:
2873:
2866:
2859:
2788:
2781:
2774:
2767:
2760:
2741:
2734:
2727:
2720:
2713:
2706:
2699:
2692:
2685:
2661:
2654:
2647:
2640:
2633:
2626:
2619:
2612:
2605:
2589:
2582:
2575:
2568:
2561:
2554:
2547:
2540:
2533:
2526:
1733:, a cover whose members are all
1029:if it has no repeated members.
928:{\displaystyle A_{4}=\{a,b,1\}.}
649:{\displaystyle S=\{a,b,c,1,2\}.}
239:A finite family of subsets of a
4390:{\displaystyle {\mathcal {F}}.}
4324:{\displaystyle {\mathcal {F}}.}
1866:{\displaystyle {\mathcal {C}}.}
1846:is contained in some member of
1162:{\displaystyle {\mathcal {F}},}
977:Any family of subsets of a set
4472:{\displaystyle {\mathcal {F}}}
4097:{\displaystyle {\mathcal {F}}}
2981:{\displaystyle A_{i}\nearrow }
2975:
2850:{\displaystyle A_{i}\nearrow }
2844:
2818:{\displaystyle A_{i}\searrow }
2812:
2184:{\displaystyle {\mathcal {F}}}
2074:{\displaystyle {\mathcal {F}}}
1839:{\displaystyle {\mathcal {F}}}
1815:{\displaystyle {\mathcal {C}}}
1785:{\displaystyle {\mathcal {F}}}
1390:
1378:
1218:{\displaystyle {\mathcal {F}}}
1078:{\displaystyle {\mathcal {F}}}
1016:
1010:
559:
553:
375:
369:
340:
334:
1:
1883:Special types of set families
1301:together with another set of
192:, known as the index set, to
5414:Basic concepts in set theory
4696:Brualdi, Richard A. (2010),
4293:{\displaystyle B\setminus A}
4234:{\displaystyle \varnothing }
2935:{\displaystyle A\subseteq B}
2304:{\displaystyle B\setminus A}
2216:{\displaystyle \,\supseteq }
1562:A family of sets is said to
1516:{\displaystyle \varnothing }
4682:, Oxford: Clarendon Press,
4035:
4028:
3969:
3962:
3955:
3936:
3929:
3870:
3863:
3856:
3840:
3745:
3666:
3647:
3609:
3602:
3582:
3575:
3568:
3538:
3531:
3503:
3496:
3489:
3470:
3463:
3456:
3449:
3442:
3435:
3428:
3421:
3414:
3392:
3385:
3364:
3357:
3350:
3343:
3336:
3314:
3300:
3293:
3279:
3272:
3265:
3258:
3239:
3218:
3204:
3197:
3190:
3183:
3164:
3129:
3122:
3115:
3108:
3040:
3033:
3026:
3004:
2997:
2945:
2895:
2742:
2693:
2686:
2662:
2613:
2606:
2534:
2527:
1924:abstract simplicial complex
1312:abstract simplicial complex
1085:is any family of sets then
5435:
5264:von Neumann–Bernays–Gödel
4698:Introductory Combinatorics
4455:are arbitrary elements of
4257:
4021:
3983:
3976:
3891:
3884:
3877:
3802:
3795:
3788:
3775:
3768:
3761:
3707:
3700:
3693:
3680:
3673:
3595:
3552:
3545:
3524:
3517:
3510:
3378:
3371:
3307:
3286:
3232:
3225:
3211:
3157:
3150:
3143:
3136:
3089:
3082:
3075:
3068:
3061:
3054:
3047:
2952:
2909:
2902:
2874:
2867:
2860:
2789:
2782:
2775:
2768:
2761:
2735:
2728:
2721:
2714:
2707:
2700:
2655:
2648:
2641:
2634:
2627:
2620:
2590:
2583:
2576:
2569:
2562:
2555:
2548:
2541:
2054:
1551:
997:is itself a subset of the
5065:One-to-one correspondence
4802:
4678:Biggs, Norman L. (1985),
4650:Roberts & Tesman 2009
1396:{\displaystyle (X,\tau )}
1225:of sets is a family over
408:is a family of sets over
4571:λ-system (Dynkin system)
1798:another (coarser) cover
1654:. A family is called a
1628:that is also a cover of
594:form a family of sets.
349:{\displaystyle \wp (S).}
34:and related branches of
4479:and it is assumed that
4274:where every complement
4208:{\displaystyle \Omega }
4164:{\displaystyle \Omega }
4048:Is necessarily true of
2275:{\displaystyle A\cup B}
2246:{\displaystyle A\cap B}
2135:Is necessarily true of
2096:{\displaystyle \Omega }
1657:point-finite collection
1500:that contains both the
1360:, it can be used as an
1048:Hall's marriage theorem
1022:{\displaystyle \wp (S)}
567:{\displaystyle S^{(k)}}
381:{\displaystyle \wp (S)}
5023:Constructible universe
4850:Constructibility (V=L)
4559:Generalized quantifier
4506:
4473:
4449:
4391:
4360:
4325:
4294:
4235:
4209:
4165:
4098:
4069:
4010:
3918:
3831:
3736:
3638:
2982:
2936:
2851:
2819:
2498:
2465:
2432:
2383:
2334:
2305:
2276:
2247:
2217:
2185:
2156:
2097:
2075:
1988:
1965:
1941:
1867:
1840:
1816:
1786:
1743:. A family is called
1723:
1701:
1674:
1642:
1622:
1602:
1582:
1537:
1517:
1494:
1470:
1441:
1417:
1397:
1276:
1246:
1219:
1195:
1163:
1136:
1079:
1023:
991:
951:
929:
875:
759:
670:
650:
588:
568:
534:
513:
488:
467:
447:
425:
402:
382:
350:
318:
294:
256:
226:
206:
186:
154:
127:
103:
76:
5246:Principia Mathematica
5080:Transfinite induction
4939:(i.e. set difference)
4716:Applied Combinatorics
4507:
4474:
4450:
4392:
4366:is equal to a finite
4361:
4326:
4300:is equal to a finite
4295:
4236:
4210:
4166:
4099:
4070:
4011:
4009:{\displaystyle \cap }
3919:
3917:{\displaystyle \cup }
3832:
3737:
3639:
2983:
2937:
2852:
2820:
2499:
2466:
2433:
2384:
2335:
2306:
2277:
2248:
2218:
2186:
2157:
2098:
2076:
2026:(with respect to the
2013:is a family of sets.
2003:augmentation property
1989:
1966:
1942:
1868:
1841:
1817:
1787:
1724:
1702:
1675:
1643:
1623:
1603:
1583:
1548:Covers and topologies
1538:
1518:
1495:
1471:
1442:
1440:{\displaystyle \tau }
1418:
1398:
1362:error-correcting code
1337:, and an (arbitrary)
1329:consists of a set of
1277:
1247:
1220:
1196:
1169:where in particular,
1164:
1137:
1080:
1024:
992:
952:
930:
876:
760:
671:
651:
589:
569:
535:
514:
489:
473:elements is called a
468:
448:
426:
403:
383:
351:
319:
295:
257:
227:
207:
187:
155:
128:
104:
77:
5320:Burali-Forti paradox
5075:Set-builder notation
5028:Continuum hypothesis
4968:Symmetric difference
4747:at Wikimedia Commons
4680:Discrete Mathematics
4541:Combinatorial design
4483:
4459:
4401:
4374:
4344:
4308:
4278:
4225:
4199:
4155:
4084:
4052:
4000:
3908:
3811:
3716:
3618:
2962:
2920:
2831:
2799:
2478:
2445:
2396:
2347:
2318:
2289:
2260:
2231:
2206:
2171:
2139:
2087:
2061:
2035:independence systems
1975:
1955:
1931:
1850:
1826:
1802:
1772:
1713:
1688:
1664:
1632:
1612:
1592:
1572:
1527:
1507:
1484:
1457:
1431:
1407:
1375:
1256:
1229:
1205:
1173:
1146:
1089:
1065:
1004:
981:
941:
885:
769:
684:
660:
601:
578:
545:
524:
500:
478:
457:
437:
412:
392:
363:
328:
308:
284:
246:
216:
196:
176:
141:
117:
93:
66:
5281:Tarski–Grothendieck
2047:bornological spaces
1822:if every member of
1558:Filters in topology
1371:consists of a pair
1327:incidence structure
680:sense) is given by
270:extremal set theory
4870:Limitation of size
4535:Class (set theory)
4502:
4469:
4445:
4387:
4356:
4321:
4290:
4231:
4205:
4161:
4094:
4065:
4006:
3914:
3827:
3732:
3634:
2978:
2932:
2847:
2815:
2494:
2461:
2428:
2379:
2330:
2301:
2272:
2243:
2213:
2181:
2152:
2093:
2071:
2028:inclusion relation
1987:{\displaystyle F.}
1984:
1961:
1937:
1863:
1836:
1812:
1782:
1719:
1700:{\displaystyle X.}
1697:
1670:
1660:if every point of
1638:
1618:
1598:
1588:if every point of
1578:
1533:
1513:
1490:
1469:{\displaystyle X,}
1466:
1437:
1413:
1393:
1343:incidence relation
1316:simplicial complex
1272:
1242:
1215:
1191:
1159:
1132:
1127:
1126:
1075:
1019:
987:
947:
925:
871:
755:
666:
646:
584:
564:
530:
512:{\displaystyle S.}
509:
484:
463:
443:
424:{\displaystyle S.}
421:
398:
378:
346:
324:and is denoted by
314:
290:
252:
222:
202:
182:
153:{\displaystyle S.}
150:
123:
99:
72:
5401:
5400:
5310:Russell's paradox
5259:Zermelo–Fraenkel
5160:Dedekind-infinite
5033:Diagonal argument
4932:Cartesian product
4796:Set (mathematics)
4743:Media related to
4725:978-1-4200-9982-9
4589:Russell's paradox
4517:
4516:
1964:{\displaystyle F}
1940:{\displaystyle F}
1895:Sperner's theorem
1731:topological space
1722:{\displaystyle X}
1673:{\displaystyle X}
1641:{\displaystyle X}
1621:{\displaystyle X}
1601:{\displaystyle X}
1581:{\displaystyle X}
1536:{\displaystyle X}
1493:{\displaystyle X}
1416:{\displaystyle X}
1369:topological space
1107:
1040:of all sets (the
990:{\displaystyle S}
669:{\displaystyle S}
587:{\displaystyle S}
533:{\displaystyle k}
487:{\displaystyle k}
466:{\displaystyle k}
446:{\displaystyle S}
401:{\displaystyle S}
317:{\displaystyle S}
293:{\displaystyle S}
268:. The subject of
262:is also called a
255:{\displaystyle S}
225:{\displaystyle I}
205:{\displaystyle F}
185:{\displaystyle I}
126:{\displaystyle S}
111:family of subsets
102:{\displaystyle S}
75:{\displaystyle F}
16:(Redirected from
5426:
5419:Families of sets
5383:Bertrand Russell
5373:John von Neumann
5358:Abraham Fraenkel
5353:Richard Dedekind
5315:Suslin's problem
5226:Cantor's theorem
4943:De Morgan's laws
4808:
4775:
4768:
4761:
4752:
4742:
4728:
4710:
4692:
4665:
4659:
4653:
4647:
4641:
4635:
4629:
4626:Naive Set Theory
4622:
4511:
4509:
4508:
4503:
4492:
4491:
4478:
4476:
4475:
4470:
4468:
4467:
4454:
4452:
4451:
4446:
4438:
4437:
4425:
4424:
4396:
4394:
4393:
4388:
4383:
4382:
4365:
4363:
4362:
4357:
4330:
4328:
4327:
4322:
4317:
4316:
4299:
4297:
4296:
4291:
4271:
4259:Additionally, a
4253:
4242:
4241:
4240:
4238:
4237:
4232:
4216:
4215:
4214:
4212:
4211:
4206:
4190:
4189:
4181:
4180:
4172:
4171:
4170:
4168:
4167:
4162:
4144:
4143:
4135:
4134:
4126:
4125:
4117:
4108:
4107:
4103:
4101:
4100:
4095:
4093:
4092:
4076:
4075:
4074:
4072:
4071:
4066:
4061:
4060:
4038:
4037:
4031:
4030:
4024:
4023:
4017:
4015:
4013:
4012:
4007:
3996:(even arbitrary
3993:
3986:
3985:
3979:
3978:
3972:
3971:
3965:
3964:
3958:
3957:
3950:
3939:
3938:
3932:
3931:
3925:
3923:
3921:
3920:
3915:
3904:(even arbitrary
3901:
3894:
3893:
3887:
3886:
3880:
3879:
3873:
3872:
3866:
3865:
3859:
3858:
3851:
3843:
3842:
3836:
3834:
3833:
3828:
3826:
3825:
3805:
3804:
3798:
3797:
3791:
3790:
3778:
3777:
3771:
3770:
3764:
3763:
3756:
3748:
3747:
3741:
3739:
3738:
3733:
3731:
3730:
3710:
3709:
3703:
3702:
3696:
3695:
3683:
3682:
3676:
3675:
3669:
3668:
3661:
3658:
3650:
3649:
3643:
3641:
3640:
3635:
3633:
3632:
3612:
3611:
3605:
3604:
3598:
3597:
3585:
3584:
3578:
3577:
3571:
3570:
3563:
3555:
3554:
3548:
3547:
3541:
3540:
3534:
3533:
3527:
3526:
3520:
3519:
3513:
3512:
3506:
3505:
3499:
3498:
3492:
3491:
3484:
3473:
3472:
3466:
3465:
3459:
3458:
3452:
3451:
3445:
3444:
3438:
3437:
3431:
3430:
3424:
3423:
3417:
3416:
3409:
3406:
3395:
3394:
3388:
3387:
3381:
3380:
3374:
3373:
3367:
3366:
3360:
3359:
3353:
3352:
3346:
3345:
3339:
3338:
3331:
3330:
3317:
3316:
3310:
3309:
3303:
3302:
3296:
3295:
3289:
3288:
3282:
3281:
3275:
3274:
3268:
3267:
3261:
3260:
3253:
3242:
3241:
3235:
3234:
3228:
3227:
3221:
3220:
3214:
3213:
3207:
3206:
3200:
3199:
3193:
3192:
3186:
3185:
3178:
3167:
3166:
3160:
3159:
3153:
3152:
3146:
3145:
3139:
3138:
3132:
3131:
3125:
3124:
3118:
3117:
3111:
3110:
3103:
3102:
3101:(Measure theory)
3092:
3091:
3085:
3084:
3078:
3077:
3071:
3070:
3064:
3063:
3057:
3056:
3050:
3049:
3043:
3042:
3036:
3035:
3029:
3028:
3021:
3020:
3007:
3006:
3000:
2999:
2987:
2985:
2984:
2979:
2974:
2973:
2955:
2954:
2948:
2947:
2941:
2939:
2938:
2933:
2912:
2911:
2905:
2904:
2898:
2897:
2890:
2889:
2885:
2877:
2876:
2870:
2869:
2863:
2862:
2856:
2854:
2853:
2848:
2843:
2842:
2824:
2822:
2821:
2816:
2811:
2810:
2792:
2791:
2785:
2784:
2778:
2777:
2771:
2770:
2764:
2763:
2756:
2745:
2744:
2738:
2737:
2731:
2730:
2724:
2723:
2717:
2716:
2710:
2709:
2703:
2702:
2696:
2695:
2689:
2688:
2682:
2680:
2677:
2665:
2664:
2658:
2657:
2651:
2650:
2644:
2643:
2637:
2636:
2630:
2629:
2623:
2622:
2616:
2615:
2609:
2608:
2602:
2593:
2592:
2586:
2585:
2579:
2578:
2572:
2571:
2565:
2564:
2558:
2557:
2551:
2550:
2544:
2543:
2537:
2536:
2530:
2529:
2523:
2520:
2511:
2504:
2503:
2501:
2500:
2495:
2493:
2492:
2471:
2470:
2468:
2467:
2462:
2460:
2459:
2438:
2437:
2435:
2434:
2429:
2421:
2420:
2408:
2407:
2389:
2388:
2386:
2385:
2380:
2372:
2371:
2359:
2358:
2340:
2339:
2337:
2336:
2331:
2311:
2310:
2308:
2307:
2302:
2282:
2281:
2279:
2278:
2273:
2253:
2252:
2250:
2249:
2244:
2224:
2222:
2220:
2219:
2214:
2195:
2194:
2190:
2188:
2187:
2182:
2180:
2179:
2163:
2162:
2161:
2159:
2158:
2153:
2148:
2147:
2124:
2117:
2110:
2103:
2102:
2100:
2099:
2094:
2080:
2078:
2077:
2072:
2070:
2069:
2052:
1993:
1991:
1990:
1985:
1970:
1968:
1967:
1962:
1946:
1944:
1943:
1938:
1927:is a set family
1916:Euclidean spaces
1872:
1870:
1869:
1864:
1859:
1858:
1845:
1843:
1842:
1837:
1835:
1834:
1821:
1819:
1818:
1813:
1811:
1810:
1791:
1789:
1788:
1783:
1781:
1780:
1756:σ-locally finite
1728:
1726:
1725:
1720:
1706:
1704:
1703:
1698:
1679:
1677:
1676:
1671:
1647:
1645:
1644:
1639:
1627:
1625:
1624:
1619:
1607:
1605:
1604:
1599:
1587:
1585:
1584:
1579:
1542:
1540:
1539:
1534:
1522:
1520:
1519:
1514:
1499:
1497:
1496:
1491:
1475:
1473:
1472:
1467:
1446:
1444:
1443:
1438:
1422:
1420:
1419:
1414:
1402:
1400:
1399:
1394:
1358:Hamming distance
1285:Related concepts
1281:
1279:
1278:
1273:
1268:
1267:
1251:
1249:
1248:
1243:
1241:
1240:
1224:
1222:
1221:
1216:
1214:
1213:
1200:
1198:
1197:
1192:
1168:
1166:
1165:
1160:
1155:
1154:
1141:
1139:
1138:
1133:
1128:
1125:
1124:
1123:
1101:
1100:
1084:
1082:
1081:
1076:
1074:
1073:
1028:
1026:
1025:
1020:
996:
994:
993:
988:
956:
954:
953:
948:
934:
932:
931:
926:
897:
896:
880:
878:
877:
872:
849:
848:
818:
817:
781:
780:
764:
762:
761:
756:
751:
747:
746:
745:
733:
732:
720:
719:
707:
706:
675:
673:
672:
667:
655:
653:
652:
647:
593:
591:
590:
585:
573:
571:
570:
565:
563:
562:
539:
537:
536:
531:
518:
516:
515:
510:
493:
491:
490:
485:
472:
470:
469:
464:
452:
450:
449:
444:
430:
428:
427:
422:
407:
405:
404:
399:
387:
385:
384:
379:
355:
353:
352:
347:
323:
321:
320:
315:
299:
297:
296:
291:
261:
259:
258:
253:
231:
229:
228:
223:
211:
209:
208:
203:
191:
189:
188:
183:
159:
157:
156:
151:
132:
130:
129:
124:
108:
106:
105:
100:
81:
79:
78:
73:
62:. A collection
21:
5434:
5433:
5429:
5428:
5427:
5425:
5424:
5423:
5404:
5403:
5402:
5397:
5324:
5303:
5287:
5252:New Foundations
5199:
5089:
5008:Cardinal number
4991:
4977:
4918:
4809:
4800:
4784:
4779:
4735:
4726:
4713:
4708:
4695:
4690:
4677:
4674:
4669:
4668:
4660:
4656:
4648:
4644:
4636:
4632:
4623:
4619:
4614:
4529:Algebra of sets
4525:
4519:
4512:
4481:
4480:
4457:
4456:
4429:
4416:
4399:
4398:
4397:
4372:
4371:
4342:
4341:
4331:
4306:
4305:
4276:
4275:
4269:
4250:
4248:
4245:
4223:
4222:
4220:
4219:
4197:
4196:
4194:
4193:
4187:
4185:
4184:
4178:
4176:
4175:
4153:
4152:
4150:
4148:
4147:
4141:
4139:
4138:
4132:
4130:
4129:
4123:
4121:
4120:
4114:
4111:
4082:
4081:
4079:
4078:
4077:
4050:
4049:
4047:
4046:
3998:
3997:
3995:
3994:
3949:Closed Topology
3948:
3906:
3905:
3903:
3902:
3849:
3809:
3808:
3754:
3714:
3713:
3659:
3656:
3616:
3615:
3561:
3482:
3407:
3404:
3328:
3326:
3251:
3176:
3100:
3098:
3018:
3016:
2989:
2965:
2960:
2959:
2918:
2917:
2916:
2888:(Dynkin System)
2887:
2886:
2883:
2834:
2829:
2828:
2802:
2797:
2796:
2754:
2678:
2675:
2673:
2598:
2518:
2516:
2507:
2476:
2475:
2474:
2443:
2442:
2441:
2412:
2399:
2394:
2393:
2392:
2363:
2350:
2345:
2344:
2343:
2316:
2315:
2314:
2287:
2286:
2285:
2258:
2257:
2256:
2229:
2228:
2227:
2204:
2203:
2201:
2198:
2169:
2168:
2166:
2165:
2164:
2137:
2136:
2134:
2133:
2128:
2085:
2084:
2059:
2058:
2055:
2019:convexity space
1973:
1972:
1953:
1952:
1949:downward closed
1929:
1928:
1908:Helly's theorem
1885:
1876:star refinement
1848:
1847:
1824:
1823:
1800:
1799:
1770:
1769:
1711:
1710:
1686:
1685:
1662:
1661:
1630:
1629:
1610:
1609:
1590:
1589:
1570:
1569:
1560:
1550:
1525:
1524:
1505:
1504:
1482:
1481:
1455:
1454:
1429:
1428:
1405:
1404:
1373:
1372:
1339:binary relation
1287:
1254:
1253:
1227:
1226:
1203:
1202:
1171:
1170:
1144:
1143:
1087:
1086:
1063:
1062:
1002:
1001:
979:
978:
975:
959:ordinal numbers
939:
938:
888:
883:
882:
840:
809:
772:
767:
766:
737:
724:
711:
698:
697:
693:
682:
681:
658:
657:
599:
598:
576:
575:
548:
543:
542:
522:
521:
498:
497:
476:
475:
455:
454:
435:
434:
410:
409:
390:
389:
388:of a given set
361:
360:
326:
325:
306:
305:
282:
281:
278:
244:
243:
214:
213:
194:
193:
174:
173:
139:
138:
115:
114:
91:
90:
64:
63:
28:
23:
22:
15:
12:
11:
5:
5432:
5430:
5422:
5421:
5416:
5406:
5405:
5399:
5398:
5396:
5395:
5390:
5388:Thoralf Skolem
5385:
5380:
5375:
5370:
5365:
5360:
5355:
5350:
5345:
5340:
5334:
5332:
5326:
5325:
5323:
5322:
5317:
5312:
5306:
5304:
5302:
5301:
5298:
5292:
5289:
5288:
5286:
5285:
5284:
5283:
5278:
5273:
5272:
5271:
5256:
5255:
5254:
5242:
5241:
5240:
5229:
5228:
5223:
5218:
5213:
5207:
5205:
5201:
5200:
5198:
5197:
5192:
5187:
5182:
5173:
5168:
5163:
5153:
5148:
5147:
5146:
5141:
5136:
5126:
5116:
5111:
5106:
5100:
5098:
5091:
5090:
5088:
5087:
5082:
5077:
5072:
5070:Ordinal number
5067:
5062:
5057:
5052:
5051:
5050:
5045:
5035:
5030:
5025:
5020:
5015:
5005:
5000:
4994:
4992:
4990:
4989:
4986:
4982:
4979:
4978:
4976:
4975:
4970:
4965:
4960:
4955:
4950:
4948:Disjoint union
4945:
4940:
4934:
4928:
4926:
4920:
4919:
4917:
4916:
4915:
4914:
4909:
4898:
4897:
4895:Martin's axiom
4892:
4887:
4882:
4877:
4872:
4867:
4862:
4860:Extensionality
4857:
4852:
4847:
4846:
4845:
4840:
4835:
4825:
4819:
4817:
4811:
4810:
4803:
4801:
4799:
4798:
4792:
4790:
4786:
4785:
4780:
4778:
4777:
4770:
4763:
4755:
4749:
4748:
4734:
4733:External links
4731:
4730:
4729:
4724:
4711:
4706:
4693:
4688:
4673:
4670:
4667:
4666:
4654:
4642:
4630:
4616:
4615:
4613:
4610:
4609:
4608:
4602:
4596:
4586:
4580:
4574:
4568:
4565:Indexed family
4562:
4556:
4550:
4544:
4538:
4532:
4524:
4521:
4515:
4514:
4501:
4498:
4495:
4490:
4466:
4444:
4441:
4436:
4432:
4428:
4423:
4419:
4415:
4412:
4409:
4406:
4386:
4381:
4368:disjoint union
4355:
4352:
4349:
4338:
4320:
4315:
4302:disjoint union
4289:
4286:
4283:
4265:
4255:
4254:
4243:
4230:
4217:
4204:
4191:
4182:
4173:
4160:
4145:
4136:
4127:
4118:
4109:
4091:
4064:
4059:
4043:
4042:
4039:
4032:
4025:
4018:
4005:
3987:
3980:
3973:
3966:
3959:
3952:
3944:
3943:
3940:
3933:
3926:
3913:
3895:
3888:
3881:
3874:
3867:
3860:
3853:
3845:
3844:
3837:
3824:
3819:
3816:
3806:
3799:
3792:
3785:
3782:
3779:
3772:
3765:
3758:
3755:Filter subbase
3750:
3749:
3742:
3729:
3724:
3721:
3711:
3704:
3697:
3690:
3687:
3684:
3677:
3670:
3663:
3652:
3651:
3644:
3631:
3626:
3623:
3613:
3606:
3599:
3592:
3589:
3586:
3579:
3572:
3565:
3557:
3556:
3549:
3542:
3535:
3528:
3521:
3514:
3507:
3500:
3493:
3486:
3478:
3477:
3474:
3467:
3460:
3453:
3446:
3439:
3432:
3425:
3418:
3411:
3400:
3399:
3396:
3389:
3382:
3375:
3368:
3361:
3354:
3347:
3340:
3333:
3322:
3321:
3318:
3311:
3304:
3297:
3290:
3283:
3276:
3269:
3262:
3255:
3247:
3246:
3243:
3236:
3229:
3222:
3215:
3208:
3201:
3194:
3187:
3180:
3172:
3171:
3168:
3161:
3154:
3147:
3140:
3133:
3126:
3119:
3112:
3105:
3094:
3093:
3086:
3079:
3072:
3065:
3058:
3051:
3044:
3037:
3030:
3023:
3019:(Order theory)
3012:
3011:
3008:
3001:
2994:
2977:
2972:
2968:
2956:
2949:
2942:
2931:
2928:
2925:
2913:
2906:
2899:
2892:
2879:
2878:
2871:
2864:
2857:
2846:
2841:
2837:
2825:
2814:
2809:
2805:
2793:
2786:
2779:
2772:
2765:
2758:
2755:Monotone class
2750:
2749:
2746:
2739:
2732:
2725:
2718:
2711:
2704:
2697:
2690:
2683:
2670:
2669:
2666:
2659:
2652:
2645:
2638:
2631:
2624:
2617:
2610:
2603:
2595:
2594:
2587:
2580:
2573:
2566:
2559:
2552:
2545:
2538:
2531:
2524:
2513:
2512:
2505:
2491:
2486:
2483:
2472:
2458:
2453:
2450:
2439:
2427:
2424:
2419:
2415:
2411:
2406:
2402:
2390:
2378:
2375:
2370:
2366:
2362:
2357:
2353:
2341:
2329:
2326:
2323:
2312:
2300:
2297:
2294:
2283:
2271:
2268:
2265:
2254:
2242:
2239:
2236:
2225:
2212:
2196:
2178:
2151:
2146:
2130:
2129:
2127:
2126:
2119:
2112:
2104:
2092:
2068:
1983:
1980:
1960:
1936:
1890:Sperner family
1884:
1881:
1878:
1862:
1857:
1833:
1809:
1797:
1779:
1764:
1758:
1748:
1746:locally finite
1742:
1718:
1696:
1693:
1669:
1659:
1653:
1637:
1617:
1597:
1577:
1567:
1549:
1546:
1545:
1544:
1532:
1512:
1489:
1465:
1462:
1452:
1436:
1412:
1392:
1389:
1386:
1383:
1380:
1365:
1346:
1323:
1308:
1286:
1283:
1271:
1266:
1261:
1239:
1234:
1212:
1190:
1187:
1184:
1181:
1178:
1158:
1153:
1131:
1122:
1117:
1114:
1110:
1104:
1099:
1094:
1072:
1018:
1015:
1012:
1009:
986:
974:
971:
946:
924:
921:
918:
915:
912:
909:
906:
903:
900:
895:
891:
870:
867:
864:
861:
858:
855:
852:
847:
843:
839:
836:
833:
830:
827:
824:
821:
816:
812:
808:
805:
802:
799:
796:
793:
790:
787:
784:
779:
775:
754:
750:
744:
740:
736:
731:
727:
723:
718:
714:
710:
705:
701:
696:
692:
689:
665:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
609:
606:
583:
561:
558:
555:
551:
529:
508:
505:
483:
462:
442:
420:
417:
397:
377:
374:
371:
368:
345:
342:
339:
336:
333:
313:
300:is called the
289:
277:
274:
251:
221:
201:
181:
162:family of sets
149:
146:
135:family of sets
122:
98:
71:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5431:
5420:
5417:
5415:
5412:
5411:
5409:
5394:
5393:Ernst Zermelo
5391:
5389:
5386:
5384:
5381:
5379:
5378:Willard Quine
5376:
5374:
5371:
5369:
5366:
5364:
5361:
5359:
5356:
5354:
5351:
5349:
5346:
5344:
5341:
5339:
5336:
5335:
5333:
5331:
5330:Set theorists
5327:
5321:
5318:
5316:
5313:
5311:
5308:
5307:
5305:
5299:
5297:
5294:
5293:
5290:
5282:
5279:
5277:
5276:Kripke–Platek
5274:
5270:
5267:
5266:
5265:
5262:
5261:
5260:
5257:
5253:
5250:
5249:
5248:
5247:
5243:
5239:
5236:
5235:
5234:
5231:
5230:
5227:
5224:
5222:
5219:
5217:
5214:
5212:
5209:
5208:
5206:
5202:
5196:
5193:
5191:
5188:
5186:
5183:
5181:
5179:
5174:
5172:
5169:
5167:
5164:
5161:
5157:
5154:
5152:
5149:
5145:
5142:
5140:
5137:
5135:
5132:
5131:
5130:
5127:
5124:
5120:
5117:
5115:
5112:
5110:
5107:
5105:
5102:
5101:
5099:
5096:
5092:
5086:
5083:
5081:
5078:
5076:
5073:
5071:
5068:
5066:
5063:
5061:
5058:
5056:
5053:
5049:
5046:
5044:
5041:
5040:
5039:
5036:
5034:
5031:
5029:
5026:
5024:
5021:
5019:
5016:
5013:
5009:
5006:
5004:
5001:
4999:
4996:
4995:
4993:
4987:
4984:
4983:
4980:
4974:
4971:
4969:
4966:
4964:
4961:
4959:
4956:
4954:
4951:
4949:
4946:
4944:
4941:
4938:
4935:
4933:
4930:
4929:
4927:
4925:
4921:
4913:
4912:specification
4910:
4908:
4905:
4904:
4903:
4900:
4899:
4896:
4893:
4891:
4888:
4886:
4883:
4881:
4878:
4876:
4873:
4871:
4868:
4866:
4863:
4861:
4858:
4856:
4853:
4851:
4848:
4844:
4841:
4839:
4836:
4834:
4831:
4830:
4829:
4826:
4824:
4821:
4820:
4818:
4816:
4812:
4807:
4797:
4794:
4793:
4791:
4787:
4783:
4776:
4771:
4769:
4764:
4762:
4757:
4756:
4753:
4746:
4741:
4737:
4736:
4732:
4727:
4721:
4717:
4712:
4709:
4707:0-13-602040-2
4703:
4699:
4694:
4691:
4689:0-19-853252-0
4685:
4681:
4676:
4675:
4671:
4663:
4658:
4655:
4651:
4646:
4643:
4639:
4634:
4631:
4627:
4621:
4618:
4611:
4606:
4603:
4600:
4597:
4594:
4590:
4587:
4584:
4581:
4578:
4575:
4572:
4569:
4566:
4563:
4560:
4557:
4554:
4553:Field of sets
4551:
4548:
4545:
4542:
4539:
4536:
4533:
4530:
4527:
4526:
4522:
4520:
4513:
4499:
4493:
4442:
4439:
4434:
4430:
4426:
4421:
4417:
4413:
4410:
4407:
4404:
4384:
4369:
4353:
4339:
4337:
4334:
4318:
4303:
4287:
4281:
4273:
4266:
4264:
4261:
4256:
4252:
4244:
4218:
4192:
4183:
4179:intersections
4174:
4146:
4137:
4128:
4124:intersections
4119:
4116:
4110:
4106:
4105:closed under:
4062:
4045:
4044:
4040:
4033:
4026:
4019:
4003:
3992:
3988:
3981:
3974:
3967:
3960:
3953:
3951:
3946:
3945:
3941:
3934:
3927:
3911:
3900:
3896:
3889:
3882:
3875:
3868:
3861:
3854:
3852:
3850:Open Topology
3847:
3846:
3838:
3817:
3807:
3800:
3793:
3786:
3783:
3780:
3773:
3766:
3759:
3757:
3752:
3751:
3743:
3722:
3712:
3705:
3698:
3691:
3688:
3685:
3678:
3671:
3664:
3662:
3660:(Filter base)
3654:
3653:
3645:
3624:
3614:
3607:
3600:
3593:
3590:
3587:
3580:
3573:
3566:
3564:
3559:
3558:
3550:
3543:
3536:
3529:
3522:
3515:
3508:
3501:
3494:
3487:
3485:
3480:
3479:
3475:
3468:
3461:
3454:
3447:
3440:
3433:
3426:
3419:
3412:
3410:
3402:
3401:
3397:
3390:
3383:
3376:
3369:
3362:
3355:
3348:
3341:
3334:
3332:
3324:
3323:
3319:
3312:
3305:
3298:
3291:
3284:
3277:
3270:
3263:
3256:
3254:
3249:
3248:
3244:
3237:
3230:
3223:
3216:
3209:
3202:
3195:
3188:
3181:
3179:
3174:
3173:
3169:
3162:
3155:
3148:
3141:
3134:
3127:
3120:
3113:
3106:
3104:
3096:
3095:
3087:
3080:
3073:
3066:
3059:
3052:
3045:
3038:
3031:
3024:
3022:
3014:
3013:
3009:
3002:
2995:
2993:
2970:
2966:
2957:
2950:
2943:
2929:
2926:
2923:
2914:
2907:
2900:
2893:
2891:
2881:
2880:
2872:
2865:
2858:
2839:
2835:
2826:
2807:
2803:
2794:
2787:
2780:
2773:
2766:
2759:
2757:
2752:
2751:
2747:
2740:
2733:
2726:
2719:
2712:
2705:
2698:
2691:
2684:
2681:
2672:
2671:
2667:
2660:
2653:
2646:
2639:
2632:
2625:
2618:
2611:
2604:
2601:
2597:
2596:
2588:
2581:
2574:
2567:
2560:
2553:
2546:
2539:
2532:
2525:
2522:
2515:
2514:
2510:
2506:
2484:
2473:
2451:
2440:
2425:
2422:
2417:
2413:
2409:
2404:
2400:
2391:
2376:
2373:
2368:
2364:
2360:
2355:
2351:
2342:
2327:
2313:
2298:
2292:
2284:
2269:
2266:
2263:
2255:
2240:
2237:
2234:
2226:
2223:
2210:
2197:
2193:
2192:closed under:
2149:
2132:
2131:
2125:
2120:
2118:
2113:
2111:
2106:
2082:
2053:
2050:
2048:
2044:
2040:
2036:
2031:
2029:
2025:
2021:
2020:
2014:
2012:
2007:
2005:
2004:
1999:
1998:
1981:
1978:
1958:
1950:
1934:
1926:
1925:
1919:
1917:
1913:
1909:
1905:
1904:
1898:
1896:
1892:
1891:
1882:
1880:
1877:
1874:
1860:
1796:
1793:
1766:
1763:
1760:
1757:
1754:
1752:
1747:
1744:
1741:
1738:
1737:is called an
1736:
1732:
1716:
1707:
1694:
1691:
1683:
1667:
1658:
1655:
1652:
1649:
1635:
1615:
1595:
1575:
1566:
1563:
1559:
1555:
1547:
1530:
1503:
1487:
1479:
1463:
1460:
1451:
1448:
1434:
1426:
1410:
1387:
1384:
1381:
1370:
1366:
1363:
1359:
1355:
1351:
1347:
1344:
1341:, called the
1340:
1336:
1332:
1328:
1324:
1321:
1317:
1313:
1309:
1306:
1305:
1300:
1296:
1292:
1291:
1290:
1284:
1282:
1269:
1259:
1232:
1188:
1182:
1176:
1156:
1129:
1115:
1112:
1102:
1092:
1059:
1057:
1053:
1049:
1045:
1043:
1039:
1035:
1030:
1013:
1000:
984:
972:
970:
968:
964:
960:
944:
935:
922:
916:
913:
910:
907:
904:
898:
893:
889:
868:
862:
859:
856:
850:
845:
841:
837:
831:
828:
825:
819:
814:
810:
806:
800:
797:
794:
791:
788:
782:
777:
773:
752:
748:
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734:
729:
725:
721:
716:
712:
708:
703:
699:
694:
690:
687:
679:
663:
643:
637:
634:
631:
628:
625:
622:
619:
616:
613:
607:
604:
595:
581:
556:
549:
541:
527:
506:
503:
495:
481:
460:
440:
431:
418:
415:
395:
372:
359:
343:
337:
311:
303:
287:
275:
273:
271:
267:
266:
249:
242:
237:
235:
219:
199:
179:
171:
167:
163:
147:
144:
136:
120:
112:
96:
89:
85:
69:
61:
57:
53:
49:
45:
41:
37:
33:
19:
5343:Georg Cantor
5338:Paul Bernays
5269:Morse–Kelley
5244:
5177:
5176:Subset
5123:hereditarily
5085:Venn diagram
5054:
5043:ordered pair
4958:Intersection
4902:Axiom schema
4745:Set families
4715:
4697:
4679:
4657:
4645:
4638:Brualdi 2010
4633:
4625:
4620:
4592:
4583:Ring of sets
4518:
4333:
4260:
4258:
4249:Intersection
2056:
2043:antimatroids
2032:
2017:
2015:
2008:
2001:
1995:
1922:
1920:
1910:states that
1903:Helly family
1901:
1899:
1888:
1886:
1767:
1751:neighborhood
1708:
1648:is called a
1561:
1477:
1424:
1342:
1334:
1330:
1302:
1288:
1060:
1046:
1038:proper class
1031:
976:
967:proper class
962:
936:
596:
433:A subset of
432:
279:
263:
238:
234:proper class
169:
165:
161:
134:
110:
109:is called a
43:
39:
29:
5368:Thomas Jech
5211:Alternative
5190:Uncountable
5144:Ultrafilter
5003:Cardinality
4907:replacement
4855:Determinacy
4624:P. Halmos,
4370:of sets in
4336:semialgebra
4304:of sets in
4149:complements
4142:complements
2679:(Semifield)
2676:Semialgebra
1971:is also in
1912:convex sets
1792:is said to
1333:, a set of
1201:Any family
1052:Philip Hall
86:of a given
52:indexed set
36:mathematics
5408:Categories
5363:Kurt Gödel
5348:Paul Cohen
5185:Transitive
4953:Identities
4937:Complement
4924:Operations
4885:Regularity
4823:Adjunction
4782:Set theory
4672:References
4662:Biggs 1985
3483:Dual ideal
3408:(𝜎-Field)
3405:𝜎-Algebra
1740:open cover
1552:See also:
1350:block code
1304:hyperedges
1295:hypergraph
973:Properties
937:The class
265:hypergraph
241:finite set
170:set system
166:set family
44:collection
32:set theory
18:Set system
5296:Paradoxes
5216:Axiomatic
5195:Universal
5171:Singleton
5166:Recursive
5109:Countable
5104:Amorphous
4963:Power set
4880:Power set
4838:dependent
4833:countable
4652:, pg. 692
4640:, pg. 322
4599:σ-algebra
4497:∅
4494:≠
4443:…
4351:∖
4348:Ω
4285:∖
4229:∅
4221:contains
4203:Ω
4195:contains
4186:countable
4177:countable
4159:Ω
4063::
4004:∩
3912:∪
3815:∅
3720:∅
3657:Prefilter
3622:∅
2990:they are
2976:↗
2927:⊆
2884:𝜆-system
2845:↗
2813:↘
2485:∈
2482:∅
2452:∈
2449:Ω
2426:⋯
2423:∪
2410:∪
2377:⋯
2374:∩
2361:∩
2325:∖
2322:Ω
2296:∖
2267:∪
2238:∩
2211:⊇
2150::
2091:Ω
2057:Families
2039:greedoids
1735:open sets
1682:partition
1511:∅
1502:empty set
1478:open sets
1435:τ
1388:τ
1348:A binary
1320:simplices
1260:∪
1233:∪
1186:∅
1180:∅
1177:∪
1116:∈
1109:⋃
1093:∪
1050:, due to
1008:℘
999:power set
574:of a set
367:℘
358:power set
332:℘
302:power set
5300:Problems
5204:Theories
5180:Superset
5156:Infinite
4985:Concepts
4865:Infinity
4789:Overview
4664:, pg. 89
4577:π-system
4523:See also
4263:semiring
4251:Property
4140:relative
4115:downward
4113:directed
3818:∉
3723:∉
3625:∉
3327:Algebra
2992:disjoint
2958:only if
2827:only if
2795:only if
2600:Semiring
2200:Directed
1768:A cover
1651:subcover
1450:topology
1299:vertices
1042:universe
1034:subclass
678:multiset
676:(in the
540:-subsets
276:Examples
56:multiset
5238:General
5233:Zermelo
5139:subbase
5121: (
5060:Forcing
5038:Element
5010: (
4988:Methods
4875:Pairing
4272:-system
4080:or, is
3329:(Field)
3252:𝜎-Ring
2915:only if
2521:-system
2167:or, is
2081:of sets
1997:matroid
1480:) over
1036:of the
957:of all
494:-subset
453:having
168:, or a
133:, or a
84:subsets
5129:Filter
5119:Finite
5055:Family
4998:Almost
4843:global
4828:Choice
4815:Axioms
4722:
4704:
4686:
4605:σ-ring
4547:δ-ring
4247:Finite
4188:unions
4133:unions
4131:finite
4122:finite
4041:Never
3942:Never
3562:Filter
3476:Never
3398:Never
3320:Never
3245:Never
3177:δ-Ring
3170:Never
3010:Never
2748:Never
2668:Never
2509:F.I.P.
2045:, and
2024:chains
2011:filter
2009:Every
1795:refine
1568:a set
1556:, and
1427:) and
1425:points
1403:where
1354:string
1331:points
765:where
40:family
5221:Naive
5151:Fuzzy
5114:Empty
5097:types
5048:tuple
5018:Class
5012:large
4973:Union
4890:Union
4612:Notes
4267:is a
3784:Never
3781:Never
3689:Never
3686:Never
3591:Never
3588:Never
3099:Ring
3017:Ring
2083:over
1729:is a
1709:When
1565:cover
1447:is a
1335:lines
963:large
961:is a
137:over
60:class
58:, or
5134:base
4720:ISBN
4702:ISBN
4684:ISBN
1523:and
1044:).
881:and
597:Let
519:The
356:The
42:(or
38:, a
5095:Set
4151:in
2202:by
2030:).
2006:.
1921:An
1914:in
1759:or
1684:of
1453:on
1325:An
1310:An
1061:If
945:Ord
496:of
304:of
113:of
88:set
82:of
48:set
30:In
5410::
4332:A
2988:or
2049:.
2041:,
2037:,
2016:A
1994:A
1900:A
1887:A
1873:A
1367:A
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1103::=
1058:.
969:.
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164:,
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5178:·
5162:)
5158:(
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4774:e
4767:t
4760:v
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4500:.
4489:F
4465:F
4440:,
4435:2
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4427:,
4422:1
4418:A
4414:,
4411:B
4408:,
4405:A
4385:.
4380:F
4354:A
4319:.
4314:F
4288:A
4282:B
4270:π
4090:F
4058:F
4016:)
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3823:F
3728:F
3630:F
2971:i
2967:A
2930:B
2924:A
2840:i
2836:A
2808:i
2804:A
2519:π
2490:F
2457:F
2418:2
2414:A
2405:1
2401:A
2369:2
2365:A
2356:1
2352:A
2328:A
2299:A
2293:B
2270:B
2264:A
2241:B
2235:A
2177:F
2145:F
2123:e
2116:t
2109:v
2067:F
1982:.
1979:F
1959:F
1935:F
1861:.
1856:C
1832:F
1808:C
1778:F
1717:X
1695:.
1692:X
1668:X
1636:X
1616:X
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1576:X
1531:X
1488:X
1464:,
1461:X
1411:X
1391:)
1385:,
1382:X
1379:(
1270:.
1265:F
1238:F
1211:F
1189:.
1183:=
1157:,
1152:F
1130:F
1121:F
1113:F
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1071:F
1017:)
1014:S
1011:(
985:S
923:.
920:}
917:1
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911:b
908:,
905:a
902:{
899:=
894:4
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869:,
866:}
863:2
860:,
857:1
854:{
851:=
846:3
842:A
838:,
835:}
832:2
829:,
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820:=
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811:A
807:,
804:}
801:c
798:,
795:b
792:,
789:a
786:{
783:=
778:1
774:A
753:,
749:}
743:4
739:A
735:,
730:3
726:A
722:,
717:2
713:A
709:,
704:1
700:A
695:{
691:=
688:F
664:S
644:.
641:}
638:2
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632:1
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626:c
623:,
620:b
617:,
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611:{
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560:)
557:k
554:(
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416:S
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370:(
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335:(
312:S
288:S
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220:I
200:F
180:I
148:.
145:S
121:S
97:S
70:F
20:)
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