5890:
52:
36:
3542:
1357:
229:
1180:
964:
736:
663:
2300:
1241:
2378:
2149:
2071:
1993:
359:
1033:
479:
2883:
1087:
2207:
1778:
1700:
1419:
1862:
1616:
827:
2450:
2588:
901:
864:
784:
2522:
2486:
2414:
433:
2654:
2622:
392:
108:
4269:
210:
3130:
1828:
1802:
1460:
1093:
2801:
2752:
2720:
2550:
907:
4944:
4006:
1530:
1493:
2979:
3025:
3002:
2953:
2930:
3150:
3045:
2903:
2821:
2772:
2688:
669:
596:
5027:
4168:
292:
3279:
The set in which the complement is considered is thus implicitly mentioned in an absolute complement, and explicitly mentioned in a relative complement.
2216:
3079:
5341:
3331:
1191:
5914:
5499:
3364:
2306:
2077:
1999:
1921:
4287:
2826:
5354:
4677:
3695:
3508:
3190:
1878:
1562:
4939:
4023:
5359:
5349:
5086:
4292:
4837:
4283:
987:
5495:
3424:
3390:
437:
5592:
5336:
4161:
4897:
4590:
4331:
4001:
3595:
5853:
5555:
5318:
5313:
5138:
4559:
4243:
5848:
5631:
5548:
5261:
5192:
5069:
4311:
3775:
3654:
1044:
4919:
4018:
2156:
1706:
1628:
1379:
5773:
5599:
5285:
4518:
1837:
790:
4924:
4011:
2420:
5919:
5256:
4995:
4253:
4154:
3649:
3612:
2555:
5651:
5646:
5580:
5170:
4564:
4532:
4223:
4297:
870:
833:
5870:
5819:
5716:
5214:
5175:
4652:
3700:
3585:
3573:
3568:
3184:
5711:
4326:
277:, either because it has been previously specified, or it is obvious and unique, then the absolute complement of
5641:
5180:
5032:
5015:
4738:
4218:
3501:
3051:
978:
747:
2492:
2456:
2384:
2627:
2595:
5543:
5520:
5481:
5367:
5308:
4954:
4874:
4718:
4275:
4120:
4038:
3913:
3865:
3602:
3063:
5833:
5560:
5538:
5505:
5398:
5244:
5229:
5202:
5153:
5037:
4972:
4797:
4763:
4758:
4632:
4463:
4440:
4072:
3953:
3765:
3578:
1910:
548:
506:
397:
5763:
5616:
5408:
5126:
4862:
4768:
4627:
4612:
4493:
4468:
3988:
3902:
3822:
3802:
3780:
3067:
521:
133:
129:
118:
5889:
3112:
is available in the amssymb package, but this symbol is not included separately in
Unicode. The symbol
370:
86:
5736:
5698:
5575:
5379:
5219:
5143:
5121:
4949:
4907:
4806:
4773:
4637:
4425:
4336:
4062:
4052:
3886:
3817:
3770:
3710:
3590:
3202:
3329:
1175:{\displaystyle (A\setminus B)^{\complement }=A^{\complement }\cup B=A^{\complement }\cup (B\cap A).}
5865:
5756:
5741:
5721:
5678:
5565:
5515:
5441:
5386:
5323:
5116:
5111:
5059:
4827:
4816:
4488:
4388:
4316:
4307:
4303:
4238:
4233:
4057:
3968:
3881:
3876:
3871:
3685:
3627:
3558:
3494:
3104:, except that it has a little more space in front and behind the slash, akin to the LaTeX sequence
3047:
then corresponds to switching all 1s to 0s, and 0s to 1s for the logical matrix of the complement.
1500:
959:{\displaystyle {\text{If }}A\subseteq B{\text{, then }}B^{\complement }\subseteq A^{\complement }.}
587:
186:
3115:
1811:
1785:
1439:
5894:
5663:
5626:
5611:
5604:
5587:
5373:
5239:
5165:
5148:
5101:
4914:
4823:
4657:
4642:
4602:
4554:
4539:
4527:
4483:
4458:
4228:
4177:
3980:
3975:
3760:
3715:
3622:
3208:
1496:
1275:
533:
514:
5391:
4847:
2777:
2728:
2696:
2529:
5829:
5636:
5446:
5436:
5328:
5209:
5044:
5020:
4801:
4785:
4690:
4667:
4544:
4513:
4478:
4373:
4208:
3837:
3674:
3666:
3637:
3607:
3531:
3469:
3450:
3420:
3386:
3360:
3256:
3055:
2209:
demonstrating that intersection can be expressed using only the relative complement operation.
1865:
77:
5843:
5838:
5731:
5688:
5510:
5471:
5466:
5451:
5277:
5234:
5131:
4929:
4879:
4453:
4415:
4125:
4115:
4100:
4095:
3963:
3617:
3430:
3396:
3352:
3229:
3196:
5824:
5814:
5768:
5751:
5706:
5668:
5570:
5490:
5297:
5224:
5197:
5185:
5091:
5005:
4979:
4934:
4902:
4703:
4505:
4448:
4398:
4363:
4321:
3994:
3932:
3750:
3563:
3434:
3400:
3382:
3335:
3178:
3059:
2691:
2668:
1882:
1831:
1506:
1469:
731:{\displaystyle \left(A\cap B\right)^{\complement }=A^{\complement }\cup B^{\complement }.}
658:{\displaystyle \left(A\cup B\right)^{\complement }=A^{\complement }\cap B^{\complement }.}
2958:
3007:
2984:
2935:
2912:
5809:
5788:
5746:
5726:
5621:
5476:
5074:
5064:
5054:
5049:
4983:
4857:
4733:
4622:
4617:
4595:
4196:
4130:
3927:
3908:
3812:
3797:
3754:
3690:
3632:
3135:
3030:
2906:
2888:
2806:
2757:
2673:
969:
5908:
5783:
5461:
4968:
4753:
4743:
4713:
4698:
4368:
4135:
4105:
3937:
3851:
3846:
3413:
1253:
560:
517:
to 1 or 2 modulo 3 (or, in simpler terms, the integers that are not multiples of 3).
273:
be a set that contains all the elements under study; if there is no need to mention
5683:
5530:
5431:
5423:
5303:
5251:
5160:
5096:
5079:
5010:
4869:
4728:
4430:
4213:
4085:
4080:
3898:
3827:
3785:
3644:
3541:
3472:
3374:
1356:
556:
584:. The following identities capture important properties of absolute complements:
5793:
5673:
4852:
4842:
4789:
4473:
4393:
4378:
4258:
4203:
4110:
3745:
3453:
3408:
1805:
2295:{\displaystyle (B\setminus A)\cap C=(B\cap C)\setminus A=B\cap (C\setminus A).}
555:, the absolute complement of a set is generally not itself a set, but rather a
4723:
4578:
4549:
4355:
4090:
3958:
3861:
3517:
552:
69:
3092:
is usually used for rendering a set difference symbol, which is similar to a
5875:
5778:
4831:
4748:
4708:
4672:
4608:
4420:
4410:
4383:
3893:
3856:
3807:
3705:
3477:
3458:
3419:. The University Series in Undergraduate Mathematics. van Nostrand Company.
3093:
1463:
5860:
5658:
5106:
4811:
4405:
17:
1236:{\displaystyle A^{\complement }\setminus B^{\complement }=B\setminus A.}
540:
is the union of the suits of clubs and diamonds, then the complement of
5456:
4248:
2373:{\displaystyle (B\setminus A)\cup C=(B\cup C)\setminus (A\setminus C).}
2144:{\displaystyle C\setminus (B\setminus A)=(C\cap A)\cup (C\setminus B),}
2066:{\displaystyle C\setminus (A\cup B)=(C\setminus A)\cap (C\setminus B).}
1988:{\displaystyle C\setminus (A\cap B)=(C\setminus A)\cup (C\setminus B).}
490:
269:(within a larger set that is implicitly defined). In other words, let
4146:
3918:
3740:
354:{\displaystyle A^{\complement }=U\setminus A=\{x\in U:x\notin A\}.}
5000:
4346:
4191:
3790:
3550:
3257:"Complement (set) Definition (Illustrated Mathematics Dictionary)"
3085:
1495:
but this notation is ambiguous, as in some contexts (for example,
228:
51:
35:
4150:
3490:
1028:{\displaystyle \left(A^{\complement }\right)^{\complement }=A.}
474:{\displaystyle \complement _{U}A,{\text{ and }}\complement A.}
132:, i.e. all elements under consideration, are considered to be
3486:
968:(this follows from the equivalence of a conditional with its
1038:
Relationships between relative and absolute complements:
536:
of the suits of clubs, diamonds, and hearts. If the set
2878:{\displaystyle {\bar {R}}\ =\ (X\times Y)\setminus R.}
3138:
3118:
3033:
3010:
2987:
2961:
2938:
2915:
2891:
2829:
2809:
2780:
2760:
2731:
2699:
2676:
2630:
2598:
2558:
2532:
2495:
2459:
2423:
2387:
2309:
2219:
2159:
2080:
2002:
1924:
1840:
1814:
1788:
1709:
1631:
1565:
1509:
1472:
1442:
1382:
1194:
1096:
1047:
990:
910:
873:
836:
793:
750:
672:
599:
440:
400:
373:
295:
189:
89:
1913:
capture notable properties of relative complements:
1082:{\displaystyle A\setminus B=A\cap B^{\complement }.}
5802:
5697:
5529:
5422:
5274:
4967:
4890:
4784:
4688:
4577:
4504:
4439:
4354:
4345:
4267:
4184:
4071:
4034:
3946:
3836:
3724:
3665:
3549:
3524:
3181: â Identities and relationships involving sets
2202:{\displaystyle C\setminus (C\setminus A)=(C\cap A)}
1773:{\displaystyle \{2,3,4\}\setminus \{1,2,3\}=\{4\}.}
1695:{\displaystyle \{1,2,3\}\setminus \{2,3,4\}=\{1\}.}
1503:) it can be interpreted as the set of all elements
1414:{\displaystyle B\cap A^{\complement }=B\setminus A}
3412:
3187: â Set of elements common to all of some sets
3144:
3124:
3039:
3019:
2996:
2973:
2947:
2924:
2897:
2877:
2815:
2795:
2766:
2746:
2714:
2682:
2648:
2616:
2582:
2544:
2516:
2480:
2444:
2408:
2372:
2294:
2201:
2143:
2065:
1987:
1857:{\displaystyle \mathbb {R} \setminus \mathbb {Q} }
1856:
1822:
1796:
1772:
1694:
1611:{\displaystyle B\setminus A=\{x\in B:x\notin A\}.}
1610:
1524:
1487:
1454:
1413:
1235:
1174:
1081:
1027:
958:
895:
858:
822:{\displaystyle A\cap A^{\complement }=\emptyset .}
821:
778:
730:
657:
497:is the set of odd numbers, then the complement of
473:
427:
386:
353:
204:
102:
2445:{\displaystyle \emptyset \setminus A=\emptyset .}
1246:The first two complement laws above show that if
2583:{\displaystyle C\setminus A\supset C\setminus B}
544:is the union of the suits of hearts and spades.
528:is the suit of spades, then the complement of
4162:
3502:
3301:
3299:
3297:
8:
1764:
1758:
1752:
1734:
1728:
1710:
1686:
1680:
1674:
1656:
1650:
1632:
1602:
1578:
896:{\displaystyle U^{\complement }=\emptyset .}
859:{\displaystyle \emptyset ^{\complement }=U.}
345:
321:
3211: â Set of elements in any of some sets
3205: â Elements in exactly one of two sets
3193: â Equalities for combinations of sets
4988:
4583:
4351:
4169:
4155:
4147:
3509:
3495:
3487:
3137:
3117:
3032:
3009:
2986:
2960:
2937:
2914:
2890:
2831:
2830:
2828:
2808:
2779:
2759:
2733:
2732:
2730:
2698:
2675:
2629:
2597:
2557:
2531:
2494:
2458:
2422:
2386:
2308:
2218:
2158:
2079:
2001:
1923:
1850:
1849:
1842:
1841:
1839:
1816:
1815:
1813:
1790:
1789:
1787:
1708:
1630:
1564:
1508:
1471:
1441:
1393:
1381:
1212:
1199:
1193:
1145:
1126:
1113:
1095:
1070:
1046:
1010:
1000:
989:
947:
934:
925:
911:
909:
878:
872:
841:
835:
804:
792:
779:{\displaystyle A\cup A^{\complement }=U.}
761:
749:
719:
706:
693:
671:
646:
633:
620:
598:
457:
445:
439:
401:
399:
378:
372:
300:
294:
188:
94:
88:
27:Set of the elements not in a given subset
3288:
3156:. (It corresponds to the Unicode symbol
3027:Producing the complementary relation to
2517:{\displaystyle A\setminus U=\emptyset .}
2481:{\displaystyle A\setminus \emptyset =A.}
2409:{\displaystyle A\setminus A=\emptyset .}
1355:
227:
3379:Fundamentals of contemporary set theory
3221:
3080:List of mathematical symbols by subject
2909:with rows representing the elements of
2866:
2649:{\displaystyle C\supseteq B\setminus A}
2640:
2617:{\displaystyle A\supseteq B\setminus C}
2608:
2574:
2562:
2499:
2463:
2427:
2391:
2358:
2349:
2316:
2280:
2259:
2226:
2172:
2163:
2129:
2093:
2084:
2051:
2033:
2006:
1973:
1955:
1928:
1846:
1731:
1653:
1569:
1446:
1405:
1224:
1205:
1103:
1051:
489:Assume that the universe is the set of
312:
193:
3317:
3305:
45:is the area colored red in this imageâŠ
7:
3251:
3249:
3191:List of set identities and relations
1879:List of set identities and relations
1185:Relationship with a set difference:
3338:The Comprehensive LaTeX Symbol List
428:{\displaystyle {\overline {A}},A',}
236:of the white disc is the red region
3088:typesetting language, the command
3058:, complementary relations and the
2508:
2466:
2436:
2424:
2400:
887:
838:
813:
25:
5888:
3540:
2153:with the important special case
520:Assume that the universe is the
387:{\displaystyle A^{\complement }}
265:) is the set of elements not in
103:{\displaystyle A^{\complement }}
50:
34:
3230:"Complement and Set Difference"
501:is the set of even numbers. If
2863:
2851:
2836:
2738:
2364:
2352:
2346:
2334:
2322:
2310:
2286:
2274:
2256:
2244:
2232:
2220:
2196:
2184:
2178:
2166:
2135:
2123:
2117:
2105:
2099:
2087:
2057:
2045:
2039:
2027:
2021:
2009:
1979:
1967:
1961:
1949:
1943:
1931:
1166:
1154:
1110:
1097:
281:is the relative complement of
1:
5849:History of mathematical logic
3359:(in French). Paris: Hermann.
3199: â Informal set theories
509:of 3, then the complement of
205:{\displaystyle B\setminus A,}
5915:Basic concepts in set theory
5774:Primitive recursive function
3125:{\displaystyle \complement }
2690:is defined as a subset of a
1905:be three sets in a universe
1823:{\displaystyle \mathbb {Q} }
1797:{\displaystyle \mathbb {R} }
1455:{\displaystyle B\setminus A}
1341:, is the set of elements in
406:
3100:command looks identical to
3096:symbol. When rendered, the
2803:The complement of relation
1424:The relative complement of
363:The absolute complement of
5936:
4838:SchröderâBernstein theorem
4565:Monadic predicate calculus
4224:Foundations of mathematics
4007:von NeumannâBernaysâGödel
3077:
2796:{\displaystyle X\times Y.}
2747:{\displaystyle {\bar {R}}}
2715:{\displaystyle X\times Y.}
2545:{\displaystyle A\subset B}
1876:
1466:. It is sometimes written
981:or double complement law:
580:be two sets in a universe
394:. Other notations include
212:is the set of elements in
148:is the set of elements in
5884:
5871:Philosophy of mathematics
5820:Automated theorem proving
4991:
4945:Von NeumannâBernaysâGödel
4586:
3808:One-to-one correspondence
3538:
3185:Intersection (set theory)
2754:is the set complement of
547:When the universe is the
128:When all elements in the
57:⊠then the complement of
3052:composition of relations
2981:corresponds to 1 in row
2932:and columns elements of
1497:Minkowski set operations
551:described in formalized
5521:Self-verifying theories
5342:Tarski's axiomatization
4293:Tarski's undefinability
4288:incompleteness theorems
5895:Mathematics portal
5506:Proof of impossibility
5154:propositional variable
4464:Propositional calculus
3766:Constructible universe
3586:Constructibility (V=L)
3146:
3126:
3041:
3021:
2998:
2975:
2949:
2926:
2899:
2879:
2817:
2797:
2768:
2748:
2724:complementary relation
2716:
2684:
2663:Complementary relation
2650:
2618:
2584:
2546:
2518:
2482:
2446:
2410:
2374:
2296:
2203:
2145:
2067:
1989:
1858:
1824:
1798:
1774:
1696:
1612:
1526:
1489:
1456:
1421:
1415:
1237:
1176:
1083:
1029:
960:
897:
860:
823:
780:
732:
659:
559:. For more info, see
513:is the set of numbers
475:
429:
388:
367:is usually denoted by
355:
237:
206:
167:with respect to a set
104:
5764:Kolmogorov complexity
5717:Computably enumerable
5617:Model complete theory
5409:Principia Mathematica
4469:Propositional formula
4298:BanachâTarski paradox
3989:Principia Mathematica
3823:Transfinite induction
3682:(i.e. set difference)
3357:Théorie des ensembles
3147:
3127:
3068:calculus of relations
3042:
3022:
2999:
2976:
2950:
2927:
2905:is often viewed as a
2900:
2880:
2818:
2798:
2769:
2749:
2717:
2685:
2651:
2619:
2585:
2547:
2519:
2483:
2447:
2411:
2375:
2297:
2204:
2146:
2068:
1990:
1859:
1825:
1799:
1775:
1697:
1613:
1527:
1490:
1457:
1416:
1359:
1238:
1177:
1084:
1030:
961:
898:
861:
824:
781:
733:
660:
522:standard 52-card deck
476:
430:
389:
356:
231:
207:
105:
5712:ChurchâTuring thesis
5699:Computability theory
4908:continuum hypothesis
4426:Square of opposition
4284:Gödel's completeness
4063:Burali-Forti paradox
3818:Set-builder notation
3771:Continuum hypothesis
3711:Symmetric difference
3203:Symmetric difference
3136:
3116:
3106:\mathbin{\backslash}
3031:
3008:
2985:
2959:
2936:
2913:
2889:
2827:
2807:
2778:
2758:
2729:
2697:
2674:
2628:
2596:
2556:
2530:
2493:
2457:
2421:
2385:
2307:
2217:
2157:
2078:
2000:
1922:
1838:
1812:
1786:
1707:
1629:
1563:
1525:{\displaystyle b-a,}
1507:
1488:{\displaystyle B-A,}
1470:
1440:
1380:
1192:
1094:
1045:
988:
908:
871:
834:
791:
748:
670:
597:
438:
398:
371:
293:
187:
87:
5866:Mathematical object
5757:P versus NP problem
5722:Computable function
5516:Reverse mathematics
5442:Logical consequence
5319:primitive recursive
5314:elementary function
5087:Free/bound variable
4940:TarskiâGrothendieck
4459:Logical connectives
4389:Logical equivalence
4239:Logical consequence
4024:TarskiâGrothendieck
3062:are the elementary
2974:{\displaystyle aRb}
1501:functional analysis
1362:relative complement
1311:relative complement
1309:are sets, then the
1288:Relative complement
251:absolute complement
249:is a set, then the
234:absolute complement
224:Absolute complement
161:relative complement
142:absolute complement
83:, often denoted by
61:is everything else.
5920:Operations on sets
5664:Transfer principle
5627:Semantics of logic
5612:Categorical theory
5588:Non-standard model
5102:Logical connective
4229:Information theory
4178:Mathematical logic
3613:Limitation of size
3470:Weisstein, Eric W.
3451:Weisstein, Eric W.
3334:2022-03-05 at the
3261:www.mathsisfun.com
3209:Union (set theory)
3142:
3122:
3056:converse relations
3037:
3020:{\displaystyle b.}
3017:
2997:{\displaystyle a,}
2994:
2971:
2948:{\displaystyle Y.}
2945:
2925:{\displaystyle X,}
2922:
2895:
2875:
2813:
2793:
2764:
2744:
2712:
2680:
2646:
2614:
2580:
2542:
2514:
2478:
2442:
2406:
2370:
2292:
2199:
2141:
2063:
1985:
1866:irrational numbers
1854:
1820:
1794:
1770:
1692:
1608:
1522:
1485:
1464:ISO 31-11 standard
1452:
1422:
1411:
1325:, also termed the
1233:
1172:
1079:
1025:
956:
893:
856:
819:
776:
728:
655:
471:
425:
384:
351:
238:
202:
171:, also termed the
100:
5902:
5901:
5834:Abstract category
5637:Theories of truth
5447:Rule of inference
5437:Natural deduction
5418:
5417:
4963:
4962:
4668:Cartesian product
4573:
4572:
4479:Many-valued logic
4454:Boolean functions
4337:Russell's paradox
4312:diagonal argument
4209:First-order logic
4144:
4143:
4053:Russell's paradox
4002:ZermeloâFraenkel
3903:Dedekind-infinite
3776:Diagonal argument
3675:Cartesian product
3532:Set (mathematics)
3366:978-3-540-34034-8
3291:, p. E II.6.
3152:) is produced by
3145:{\displaystyle C}
3040:{\displaystyle R}
2898:{\displaystyle R}
2850:
2844:
2839:
2816:{\displaystyle R}
2767:{\displaystyle R}
2741:
2683:{\displaystyle R}
2624:is equivalent to
1462:according to the
928:
914:
741:Complement laws:
460:
409:
117:), is the set of
16:(Redirected from
5927:
5893:
5892:
5844:History of logic
5839:Category of sets
5732:Decision problem
5511:Ordinal analysis
5452:Sequent calculus
5350:Boolean algebras
5290:
5289:
5264:
5235:logical/constant
4989:
4975:
4898:ZermeloâFraenkel
4649:Set operations:
4584:
4521:
4352:
4332:LöwenheimâSkolem
4219:Formal semantics
4171:
4164:
4157:
4148:
4126:Bertrand Russell
4116:John von Neumann
4101:Abraham Fraenkel
4096:Richard Dedekind
4058:Suslin's problem
3969:Cantor's theorem
3686:De Morgan's laws
3544:
3511:
3504:
3497:
3488:
3483:
3482:
3473:"Complement Set"
3464:
3463:
3438:
3418:
3415:Naive set theory
3404:
3381:. Universitext.
3375:Devlin, Keith J.
3370:
3339:
3327:
3321:
3315:
3309:
3303:
3292:
3286:
3280:
3277:
3271:
3270:
3268:
3267:
3253:
3244:
3243:
3241:
3240:
3226:
3197:Naive set theory
3168:
3165:
3162:
3160:
3155:
3151:
3149:
3148:
3143:
3131:
3129:
3128:
3123:
3111:
3107:
3103:
3099:
3091:
3046:
3044:
3043:
3038:
3026:
3024:
3023:
3018:
3003:
3001:
3000:
2995:
2980:
2978:
2977:
2972:
2954:
2952:
2951:
2946:
2931:
2929:
2928:
2923:
2904:
2902:
2901:
2896:
2884:
2882:
2881:
2876:
2848:
2842:
2841:
2840:
2832:
2822:
2820:
2819:
2814:
2802:
2800:
2799:
2794:
2773:
2771:
2770:
2765:
2753:
2751:
2750:
2745:
2743:
2742:
2734:
2721:
2719:
2718:
2713:
2689:
2687:
2686:
2681:
2655:
2653:
2652:
2647:
2623:
2621:
2620:
2615:
2589:
2587:
2586:
2581:
2551:
2549:
2548:
2543:
2523:
2521:
2520:
2515:
2487:
2485:
2484:
2479:
2451:
2449:
2448:
2443:
2415:
2413:
2412:
2407:
2379:
2377:
2376:
2371:
2301:
2299:
2298:
2293:
2208:
2206:
2205:
2200:
2150:
2148:
2147:
2142:
2072:
2070:
2069:
2064:
1994:
1992:
1991:
1986:
1909:. The following
1908:
1904:
1898:
1892:
1863:
1861:
1860:
1855:
1853:
1845:
1832:rational numbers
1829:
1827:
1826:
1821:
1819:
1803:
1801:
1800:
1795:
1793:
1779:
1777:
1776:
1771:
1701:
1699:
1698:
1693:
1617:
1615:
1614:
1609:
1555:
1549:
1543:
1537:
1531:
1529:
1528:
1523:
1494:
1492:
1491:
1486:
1461:
1459:
1458:
1453:
1435:
1429:
1420:
1418:
1417:
1412:
1398:
1397:
1375:
1369:
1352:
1346:
1340:
1334:
1324:
1318:
1308:
1302:
1283:
1273:
1261:
1252:is a non-empty,
1251:
1242:
1240:
1239:
1234:
1217:
1216:
1204:
1203:
1181:
1179:
1178:
1173:
1150:
1149:
1131:
1130:
1118:
1117:
1088:
1086:
1085:
1080:
1075:
1074:
1034:
1032:
1031:
1026:
1015:
1014:
1009:
1005:
1004:
965:
963:
962:
957:
952:
951:
939:
938:
929:
926:
915:
912:
902:
900:
899:
894:
883:
882:
865:
863:
862:
857:
846:
845:
828:
826:
825:
820:
809:
808:
785:
783:
782:
777:
766:
765:
737:
735:
734:
729:
724:
723:
711:
710:
698:
697:
692:
688:
664:
662:
661:
656:
651:
650:
638:
637:
625:
624:
619:
615:
588:De Morgan's laws
583:
579:
575:
549:universe of sets
543:
539:
531:
527:
512:
504:
500:
496:
480:
478:
477:
472:
461:
458:
450:
449:
434:
432:
431:
426:
421:
410:
402:
393:
391:
390:
385:
383:
382:
366:
360:
358:
357:
352:
305:
304:
288:
284:
280:
276:
272:
268:
264:
256:
248:
219:
216:that are not in
215:
211:
209:
208:
203:
182:
178:
170:
166:
155:
152:that are not in
151:
147:
139:
124:
116:
109:
107:
106:
101:
99:
98:
82:
60:
54:
44:
38:
21:
5935:
5934:
5930:
5929:
5928:
5926:
5925:
5924:
5905:
5904:
5903:
5898:
5887:
5880:
5825:Category theory
5815:Algebraic logic
5798:
5769:Lambda calculus
5707:Church encoding
5693:
5669:Truth predicate
5525:
5491:Complete theory
5414:
5283:
5279:
5275:
5270:
5262:
4982: and
4978:
4973:
4959:
4935:New Foundations
4903:axiom of choice
4886:
4848:Gödel numbering
4788: and
4780:
4684:
4569:
4519:
4500:
4449:Boolean algebra
4435:
4399:Equiconsistency
4364:Classical logic
4341:
4322:Halting problem
4310: and
4286: and
4274: and
4273:
4268:Theorems (
4263:
4180:
4175:
4145:
4140:
4067:
4046:
4030:
3995:New Foundations
3942:
3832:
3751:Cardinal number
3734:
3720:
3661:
3545:
3536:
3520:
3515:
3468:
3467:
3449:
3448:
3445:
3427:
3409:Halmos, Paul R.
3407:
3393:
3373:
3367:
3351:
3348:
3343:
3342:
3336:Wayback Machine
3328:
3324:
3316:
3312:
3304:
3295:
3287:
3283:
3278:
3274:
3265:
3263:
3255:
3254:
3247:
3238:
3236:
3234:web.mnstate.edu
3228:
3227:
3223:
3218:
3179:Algebra of sets
3175:
3166:
3163:
3158:
3157:
3153:
3134:
3133:
3132:(as opposed to
3114:
3113:
3109:
3105:
3101:
3097:
3089:
3082:
3076:
3060:algebra of sets
3029:
3028:
3006:
3005:
2983:
2982:
2957:
2956:
2934:
2933:
2911:
2910:
2887:
2886:
2825:
2824:
2823:can be written
2805:
2804:
2776:
2775:
2756:
2755:
2727:
2726:
2695:
2694:
2692:product of sets
2672:
2671:
2669:binary relation
2665:
2626:
2625:
2594:
2593:
2554:
2553:
2528:
2527:
2491:
2490:
2455:
2454:
2419:
2418:
2383:
2382:
2305:
2304:
2215:
2214:
2155:
2154:
2076:
2075:
1998:
1997:
1920:
1919:
1906:
1900:
1894:
1888:
1885:
1883:Algebra of sets
1875:
1836:
1835:
1810:
1809:
1784:
1783:
1705:
1704:
1627:
1626:
1623:
1561:
1560:
1551:
1545:
1539:
1533:
1505:
1504:
1468:
1467:
1438:
1437:
1431:
1425:
1389:
1378:
1377:
1371:
1365:
1348:
1342:
1336:
1330:
1320:
1314:
1304:
1298:
1295:
1290:
1279:
1263:
1257:
1247:
1208:
1195:
1190:
1189:
1141:
1122:
1109:
1092:
1091:
1066:
1043:
1042:
996:
992:
991:
986:
985:
943:
930:
906:
905:
874:
869:
868:
837:
832:
831:
800:
789:
788:
757:
746:
745:
715:
702:
678:
674:
673:
668:
667:
642:
629:
605:
601:
600:
595:
594:
581:
577:
573:
570:
541:
537:
529:
525:
510:
502:
498:
494:
486:
459: and
441:
436:
435:
414:
396:
395:
374:
369:
368:
364:
296:
291:
290:
286:
282:
278:
274:
270:
266:
262:
257:(or simply the
254:
246:
243:
226:
217:
213:
185:
184:
180:
176:
168:
164:
153:
149:
145:
137:
136:of a given set
122:
111:
90:
85:
84:
80:
66:
65:
64:
63:
62:
58:
55:
47:
46:
42:
39:
28:
23:
22:
15:
12:
11:
5:
5933:
5931:
5923:
5922:
5917:
5907:
5906:
5900:
5899:
5885:
5882:
5881:
5879:
5878:
5873:
5868:
5863:
5858:
5857:
5856:
5846:
5841:
5836:
5827:
5822:
5817:
5812:
5810:Abstract logic
5806:
5804:
5800:
5799:
5797:
5796:
5791:
5789:Turing machine
5786:
5781:
5776:
5771:
5766:
5761:
5760:
5759:
5754:
5749:
5744:
5739:
5729:
5727:Computable set
5724:
5719:
5714:
5709:
5703:
5701:
5695:
5694:
5692:
5691:
5686:
5681:
5676:
5671:
5666:
5661:
5656:
5655:
5654:
5649:
5644:
5634:
5629:
5624:
5622:Satisfiability
5619:
5614:
5609:
5608:
5607:
5597:
5596:
5595:
5585:
5584:
5583:
5578:
5573:
5568:
5563:
5553:
5552:
5551:
5546:
5539:Interpretation
5535:
5533:
5527:
5526:
5524:
5523:
5518:
5513:
5508:
5503:
5493:
5488:
5487:
5486:
5485:
5484:
5474:
5469:
5459:
5454:
5449:
5444:
5439:
5434:
5428:
5426:
5420:
5419:
5416:
5415:
5413:
5412:
5404:
5403:
5402:
5401:
5396:
5395:
5394:
5389:
5384:
5364:
5363:
5362:
5360:minimal axioms
5357:
5346:
5345:
5344:
5333:
5332:
5331:
5326:
5321:
5316:
5311:
5306:
5293:
5291:
5272:
5271:
5269:
5268:
5267:
5266:
5254:
5249:
5248:
5247:
5242:
5237:
5232:
5222:
5217:
5212:
5207:
5206:
5205:
5200:
5190:
5189:
5188:
5183:
5178:
5173:
5163:
5158:
5157:
5156:
5151:
5146:
5136:
5135:
5134:
5129:
5124:
5119:
5114:
5109:
5099:
5094:
5089:
5084:
5083:
5082:
5077:
5072:
5067:
5057:
5052:
5050:Formation rule
5047:
5042:
5041:
5040:
5035:
5025:
5024:
5023:
5013:
5008:
5003:
4998:
4992:
4986:
4969:Formal systems
4965:
4964:
4961:
4960:
4958:
4957:
4952:
4947:
4942:
4937:
4932:
4927:
4922:
4917:
4912:
4911:
4910:
4905:
4894:
4892:
4888:
4887:
4885:
4884:
4883:
4882:
4872:
4867:
4866:
4865:
4858:Large cardinal
4855:
4850:
4845:
4840:
4835:
4821:
4820:
4819:
4814:
4809:
4794:
4792:
4782:
4781:
4779:
4778:
4777:
4776:
4771:
4766:
4756:
4751:
4746:
4741:
4736:
4731:
4726:
4721:
4716:
4711:
4706:
4701:
4695:
4693:
4686:
4685:
4683:
4682:
4681:
4680:
4675:
4670:
4665:
4660:
4655:
4647:
4646:
4645:
4640:
4630:
4625:
4623:Extensionality
4620:
4618:Ordinal number
4615:
4605:
4600:
4599:
4598:
4587:
4581:
4575:
4574:
4571:
4570:
4568:
4567:
4562:
4557:
4552:
4547:
4542:
4537:
4536:
4535:
4525:
4524:
4523:
4510:
4508:
4502:
4501:
4499:
4498:
4497:
4496:
4491:
4486:
4476:
4471:
4466:
4461:
4456:
4451:
4445:
4443:
4437:
4436:
4434:
4433:
4428:
4423:
4418:
4413:
4408:
4403:
4402:
4401:
4391:
4386:
4381:
4376:
4371:
4366:
4360:
4358:
4349:
4343:
4342:
4340:
4339:
4334:
4329:
4324:
4319:
4314:
4302:Cantor's
4300:
4295:
4290:
4280:
4278:
4265:
4264:
4262:
4261:
4256:
4251:
4246:
4241:
4236:
4231:
4226:
4221:
4216:
4211:
4206:
4201:
4200:
4199:
4188:
4186:
4182:
4181:
4176:
4174:
4173:
4166:
4159:
4151:
4142:
4141:
4139:
4138:
4133:
4131:Thoralf Skolem
4128:
4123:
4118:
4113:
4108:
4103:
4098:
4093:
4088:
4083:
4077:
4075:
4069:
4068:
4066:
4065:
4060:
4055:
4049:
4047:
4045:
4044:
4041:
4035:
4032:
4031:
4029:
4028:
4027:
4026:
4021:
4016:
4015:
4014:
3999:
3998:
3997:
3985:
3984:
3983:
3972:
3971:
3966:
3961:
3956:
3950:
3948:
3944:
3943:
3941:
3940:
3935:
3930:
3925:
3916:
3911:
3906:
3896:
3891:
3890:
3889:
3884:
3879:
3869:
3859:
3854:
3849:
3843:
3841:
3834:
3833:
3831:
3830:
3825:
3820:
3815:
3813:Ordinal number
3810:
3805:
3800:
3795:
3794:
3793:
3788:
3778:
3773:
3768:
3763:
3758:
3748:
3743:
3737:
3735:
3733:
3732:
3729:
3725:
3722:
3721:
3719:
3718:
3713:
3708:
3703:
3698:
3693:
3691:Disjoint union
3688:
3683:
3677:
3671:
3669:
3663:
3662:
3660:
3659:
3658:
3657:
3652:
3641:
3640:
3638:Martin's axiom
3635:
3630:
3625:
3620:
3615:
3610:
3605:
3603:Extensionality
3600:
3599:
3598:
3588:
3583:
3582:
3581:
3576:
3571:
3561:
3555:
3553:
3547:
3546:
3539:
3537:
3535:
3534:
3528:
3526:
3522:
3521:
3516:
3514:
3513:
3506:
3499:
3491:
3485:
3484:
3465:
3444:
3443:External links
3441:
3440:
3439:
3425:
3405:
3391:
3371:
3365:
3347:
3344:
3341:
3340:
3322:
3310:
3293:
3281:
3272:
3245:
3220:
3219:
3217:
3214:
3213:
3212:
3206:
3200:
3194:
3188:
3182:
3174:
3171:
3141:
3121:
3110:\smallsetminus
3075:
3074:LaTeX notation
3072:
3050:Together with
3036:
3016:
3013:
2993:
2990:
2970:
2967:
2964:
2944:
2941:
2921:
2918:
2907:logical matrix
2894:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2847:
2838:
2835:
2812:
2792:
2789:
2786:
2783:
2763:
2740:
2737:
2711:
2708:
2705:
2702:
2679:
2664:
2661:
2660:
2659:
2658:
2657:
2645:
2642:
2639:
2636:
2633:
2613:
2610:
2607:
2604:
2601:
2591:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2541:
2538:
2535:
2524:
2513:
2510:
2507:
2504:
2501:
2498:
2488:
2477:
2474:
2471:
2468:
2465:
2462:
2452:
2441:
2438:
2435:
2432:
2429:
2426:
2416:
2405:
2402:
2399:
2396:
2393:
2390:
2380:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2302:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2212:
2211:
2210:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2073:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
1995:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1874:
1871:
1870:
1869:
1864:is the set of
1852:
1848:
1844:
1830:is the set of
1818:
1804:is the set of
1792:
1780:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1702:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1622:
1619:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1538:is taken from
1521:
1518:
1515:
1512:
1484:
1481:
1478:
1475:
1451:
1448:
1445:
1410:
1407:
1404:
1401:
1396:
1392:
1388:
1385:
1327:set difference
1294:
1291:
1289:
1286:
1244:
1243:
1232:
1229:
1226:
1223:
1220:
1215:
1211:
1207:
1202:
1198:
1183:
1182:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1148:
1144:
1140:
1137:
1134:
1129:
1125:
1121:
1116:
1112:
1108:
1105:
1102:
1099:
1089:
1078:
1073:
1069:
1065:
1062:
1059:
1056:
1053:
1050:
1036:
1035:
1024:
1021:
1018:
1013:
1008:
1003:
999:
995:
976:
975:
974:
973:
970:contrapositive
955:
950:
946:
942:
937:
933:
924:
921:
918:
903:
892:
889:
886:
881:
877:
866:
855:
852:
849:
844:
840:
829:
818:
815:
812:
807:
803:
799:
796:
786:
775:
772:
769:
764:
760:
756:
753:
739:
738:
727:
722:
718:
714:
709:
705:
701:
696:
691:
687:
684:
681:
677:
665:
654:
649:
645:
641:
636:
632:
628:
623:
618:
614:
611:
608:
604:
569:
566:
565:
564:
545:
518:
505:is the set of
485:
482:
470:
467:
464:
456:
453:
448:
444:
424:
420:
417:
413:
408:
405:
381:
377:
350:
347:
344:
341:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
303:
299:
242:
239:
225:
222:
201:
198:
195:
192:
173:set difference
97:
93:
56:
49:
48:
40:
33:
32:
31:
30:
29:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5932:
5921:
5918:
5916:
5913:
5912:
5910:
5897:
5896:
5891:
5883:
5877:
5874:
5872:
5869:
5867:
5864:
5862:
5859:
5855:
5852:
5851:
5850:
5847:
5845:
5842:
5840:
5837:
5835:
5831:
5828:
5826:
5823:
5821:
5818:
5816:
5813:
5811:
5808:
5807:
5805:
5801:
5795:
5792:
5790:
5787:
5785:
5784:Recursive set
5782:
5780:
5777:
5775:
5772:
5770:
5767:
5765:
5762:
5758:
5755:
5753:
5750:
5748:
5745:
5743:
5740:
5738:
5735:
5734:
5733:
5730:
5728:
5725:
5723:
5720:
5718:
5715:
5713:
5710:
5708:
5705:
5704:
5702:
5700:
5696:
5690:
5687:
5685:
5682:
5680:
5677:
5675:
5672:
5670:
5667:
5665:
5662:
5660:
5657:
5653:
5650:
5648:
5645:
5643:
5640:
5639:
5638:
5635:
5633:
5630:
5628:
5625:
5623:
5620:
5618:
5615:
5613:
5610:
5606:
5603:
5602:
5601:
5598:
5594:
5593:of arithmetic
5591:
5590:
5589:
5586:
5582:
5579:
5577:
5574:
5572:
5569:
5567:
5564:
5562:
5559:
5558:
5557:
5554:
5550:
5547:
5545:
5542:
5541:
5540:
5537:
5536:
5534:
5532:
5528:
5522:
5519:
5517:
5514:
5512:
5509:
5507:
5504:
5501:
5500:from ZFC
5497:
5494:
5492:
5489:
5483:
5480:
5479:
5478:
5475:
5473:
5470:
5468:
5465:
5464:
5463:
5460:
5458:
5455:
5453:
5450:
5448:
5445:
5443:
5440:
5438:
5435:
5433:
5430:
5429:
5427:
5425:
5421:
5411:
5410:
5406:
5405:
5400:
5399:non-Euclidean
5397:
5393:
5390:
5388:
5385:
5383:
5382:
5378:
5377:
5375:
5372:
5371:
5369:
5365:
5361:
5358:
5356:
5353:
5352:
5351:
5347:
5343:
5340:
5339:
5338:
5334:
5330:
5327:
5325:
5322:
5320:
5317:
5315:
5312:
5310:
5307:
5305:
5302:
5301:
5299:
5295:
5294:
5292:
5287:
5281:
5276:Example
5273:
5265:
5260:
5259:
5258:
5255:
5253:
5250:
5246:
5243:
5241:
5238:
5236:
5233:
5231:
5228:
5227:
5226:
5223:
5221:
5218:
5216:
5213:
5211:
5208:
5204:
5201:
5199:
5196:
5195:
5194:
5191:
5187:
5184:
5182:
5179:
5177:
5174:
5172:
5169:
5168:
5167:
5164:
5162:
5159:
5155:
5152:
5150:
5147:
5145:
5142:
5141:
5140:
5137:
5133:
5130:
5128:
5125:
5123:
5120:
5118:
5115:
5113:
5110:
5108:
5105:
5104:
5103:
5100:
5098:
5095:
5093:
5090:
5088:
5085:
5081:
5078:
5076:
5073:
5071:
5068:
5066:
5063:
5062:
5061:
5058:
5056:
5053:
5051:
5048:
5046:
5043:
5039:
5036:
5034:
5033:by definition
5031:
5030:
5029:
5026:
5022:
5019:
5018:
5017:
5014:
5012:
5009:
5007:
5004:
5002:
4999:
4997:
4994:
4993:
4990:
4987:
4985:
4981:
4976:
4970:
4966:
4956:
4953:
4951:
4948:
4946:
4943:
4941:
4938:
4936:
4933:
4931:
4928:
4926:
4923:
4921:
4920:KripkeâPlatek
4918:
4916:
4913:
4909:
4906:
4904:
4901:
4900:
4899:
4896:
4895:
4893:
4889:
4881:
4878:
4877:
4876:
4873:
4871:
4868:
4864:
4861:
4860:
4859:
4856:
4854:
4851:
4849:
4846:
4844:
4841:
4839:
4836:
4833:
4829:
4825:
4822:
4818:
4815:
4813:
4810:
4808:
4805:
4804:
4803:
4799:
4796:
4795:
4793:
4791:
4787:
4783:
4775:
4772:
4770:
4767:
4765:
4764:constructible
4762:
4761:
4760:
4757:
4755:
4752:
4750:
4747:
4745:
4742:
4740:
4737:
4735:
4732:
4730:
4727:
4725:
4722:
4720:
4717:
4715:
4712:
4710:
4707:
4705:
4702:
4700:
4697:
4696:
4694:
4692:
4687:
4679:
4676:
4674:
4671:
4669:
4666:
4664:
4661:
4659:
4656:
4654:
4651:
4650:
4648:
4644:
4641:
4639:
4636:
4635:
4634:
4631:
4629:
4626:
4624:
4621:
4619:
4616:
4614:
4610:
4606:
4604:
4601:
4597:
4594:
4593:
4592:
4589:
4588:
4585:
4582:
4580:
4576:
4566:
4563:
4561:
4558:
4556:
4553:
4551:
4548:
4546:
4543:
4541:
4538:
4534:
4531:
4530:
4529:
4526:
4522:
4517:
4516:
4515:
4512:
4511:
4509:
4507:
4503:
4495:
4492:
4490:
4487:
4485:
4482:
4481:
4480:
4477:
4475:
4472:
4470:
4467:
4465:
4462:
4460:
4457:
4455:
4452:
4450:
4447:
4446:
4444:
4442:
4441:Propositional
4438:
4432:
4429:
4427:
4424:
4422:
4419:
4417:
4414:
4412:
4409:
4407:
4404:
4400:
4397:
4396:
4395:
4392:
4390:
4387:
4385:
4382:
4380:
4377:
4375:
4372:
4370:
4369:Logical truth
4367:
4365:
4362:
4361:
4359:
4357:
4353:
4350:
4348:
4344:
4338:
4335:
4333:
4330:
4328:
4325:
4323:
4320:
4318:
4315:
4313:
4309:
4305:
4301:
4299:
4296:
4294:
4291:
4289:
4285:
4282:
4281:
4279:
4277:
4271:
4266:
4260:
4257:
4255:
4252:
4250:
4247:
4245:
4242:
4240:
4237:
4235:
4232:
4230:
4227:
4225:
4222:
4220:
4217:
4215:
4212:
4210:
4207:
4205:
4202:
4198:
4195:
4194:
4193:
4190:
4189:
4187:
4183:
4179:
4172:
4167:
4165:
4160:
4158:
4153:
4152:
4149:
4137:
4136:Ernst Zermelo
4134:
4132:
4129:
4127:
4124:
4122:
4121:Willard Quine
4119:
4117:
4114:
4112:
4109:
4107:
4104:
4102:
4099:
4097:
4094:
4092:
4089:
4087:
4084:
4082:
4079:
4078:
4076:
4074:
4073:Set theorists
4070:
4064:
4061:
4059:
4056:
4054:
4051:
4050:
4048:
4042:
4040:
4037:
4036:
4033:
4025:
4022:
4020:
4019:KripkeâPlatek
4017:
4013:
4010:
4009:
4008:
4005:
4004:
4003:
4000:
3996:
3993:
3992:
3991:
3990:
3986:
3982:
3979:
3978:
3977:
3974:
3973:
3970:
3967:
3965:
3962:
3960:
3957:
3955:
3952:
3951:
3949:
3945:
3939:
3936:
3934:
3931:
3929:
3926:
3924:
3922:
3917:
3915:
3912:
3910:
3907:
3904:
3900:
3897:
3895:
3892:
3888:
3885:
3883:
3880:
3878:
3875:
3874:
3873:
3870:
3867:
3863:
3860:
3858:
3855:
3853:
3850:
3848:
3845:
3844:
3842:
3839:
3835:
3829:
3826:
3824:
3821:
3819:
3816:
3814:
3811:
3809:
3806:
3804:
3801:
3799:
3796:
3792:
3789:
3787:
3784:
3783:
3782:
3779:
3777:
3774:
3772:
3769:
3767:
3764:
3762:
3759:
3756:
3752:
3749:
3747:
3744:
3742:
3739:
3738:
3736:
3730:
3727:
3726:
3723:
3717:
3714:
3712:
3709:
3707:
3704:
3702:
3699:
3697:
3694:
3692:
3689:
3687:
3684:
3681:
3678:
3676:
3673:
3672:
3670:
3668:
3664:
3656:
3655:specification
3653:
3651:
3648:
3647:
3646:
3643:
3642:
3639:
3636:
3634:
3631:
3629:
3626:
3624:
3621:
3619:
3616:
3614:
3611:
3609:
3606:
3604:
3601:
3597:
3594:
3593:
3592:
3589:
3587:
3584:
3580:
3577:
3575:
3572:
3570:
3567:
3566:
3565:
3562:
3560:
3557:
3556:
3554:
3552:
3548:
3543:
3533:
3530:
3529:
3527:
3523:
3519:
3512:
3507:
3505:
3500:
3498:
3493:
3492:
3489:
3480:
3479:
3474:
3471:
3466:
3461:
3460:
3455:
3452:
3447:
3446:
3442:
3436:
3432:
3428:
3426:9780442030643
3422:
3417:
3416:
3410:
3406:
3402:
3398:
3394:
3392:0-387-90441-7
3388:
3384:
3380:
3376:
3372:
3368:
3362:
3358:
3354:
3350:
3349:
3345:
3337:
3333:
3330:
3326:
3323:
3319:
3314:
3311:
3308:, p. 17.
3307:
3302:
3300:
3298:
3294:
3290:
3289:Bourbaki 1970
3285:
3282:
3276:
3273:
3262:
3258:
3252:
3250:
3246:
3235:
3231:
3225:
3222:
3215:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3176:
3172:
3170:
3139:
3119:
3095:
3087:
3081:
3073:
3071:
3069:
3065:
3061:
3057:
3053:
3048:
3034:
3014:
3011:
2991:
2988:
2968:
2965:
2962:
2955:The truth of
2942:
2939:
2919:
2916:
2908:
2892:
2872:
2869:
2860:
2857:
2854:
2845:
2833:
2810:
2790:
2787:
2784:
2781:
2761:
2735:
2725:
2709:
2706:
2703:
2700:
2693:
2677:
2670:
2662:
2643:
2637:
2634:
2631:
2611:
2605:
2602:
2599:
2592:
2577:
2571:
2568:
2565:
2559:
2539:
2536:
2533:
2525:
2511:
2505:
2502:
2496:
2489:
2475:
2472:
2469:
2460:
2453:
2439:
2433:
2430:
2417:
2403:
2397:
2394:
2388:
2381:
2367:
2361:
2355:
2343:
2340:
2337:
2331:
2328:
2325:
2319:
2313:
2303:
2289:
2283:
2277:
2271:
2268:
2265:
2262:
2253:
2250:
2247:
2241:
2238:
2235:
2229:
2223:
2213:
2193:
2190:
2187:
2181:
2175:
2169:
2160:
2152:
2151:
2138:
2132:
2126:
2120:
2114:
2111:
2108:
2102:
2096:
2090:
2081:
2074:
2060:
2054:
2048:
2042:
2036:
2030:
2024:
2018:
2015:
2012:
2003:
1996:
1982:
1976:
1970:
1964:
1958:
1952:
1946:
1940:
1937:
1934:
1925:
1918:
1917:
1916:
1915:
1914:
1912:
1903:
1897:
1891:
1884:
1880:
1872:
1867:
1833:
1807:
1781:
1767:
1761:
1755:
1749:
1746:
1743:
1740:
1737:
1725:
1722:
1719:
1716:
1713:
1703:
1689:
1683:
1677:
1671:
1668:
1665:
1662:
1659:
1647:
1644:
1641:
1638:
1635:
1625:
1624:
1620:
1618:
1605:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1575:
1572:
1566:
1557:
1554:
1548:
1542:
1536:
1519:
1516:
1513:
1510:
1502:
1498:
1482:
1479:
1476:
1473:
1465:
1449:
1443:
1434:
1428:
1408:
1402:
1399:
1394:
1390:
1386:
1383:
1374:
1368:
1363:
1358:
1354:
1351:
1345:
1339:
1333:
1328:
1323:
1317:
1312:
1307:
1301:
1292:
1287:
1285:
1282:
1277:
1271:
1267:
1260:
1255:
1254:proper subset
1250:
1230:
1227:
1221:
1218:
1213:
1209:
1200:
1196:
1188:
1187:
1186:
1169:
1163:
1160:
1157:
1151:
1146:
1142:
1138:
1135:
1132:
1127:
1123:
1119:
1114:
1106:
1100:
1090:
1076:
1071:
1067:
1063:
1060:
1057:
1054:
1048:
1041:
1040:
1039:
1022:
1019:
1016:
1011:
1006:
1001:
997:
993:
984:
983:
982:
980:
971:
967:
966:
953:
948:
944:
940:
935:
931:
922:
919:
916:
904:
890:
884:
879:
875:
867:
853:
850:
847:
842:
830:
816:
810:
805:
801:
797:
794:
787:
773:
770:
767:
762:
758:
754:
751:
744:
743:
742:
725:
720:
716:
712:
707:
703:
699:
694:
689:
685:
682:
679:
675:
666:
652:
647:
643:
639:
634:
630:
626:
621:
616:
612:
609:
606:
602:
593:
592:
591:
589:
585:
567:
562:
561:universal set
558:
554:
550:
546:
535:
524:. If the set
523:
519:
516:
508:
492:
488:
487:
483:
481:
468:
465:
462:
454:
451:
446:
442:
422:
418:
415:
411:
403:
379:
375:
361:
348:
342:
339:
336:
333:
330:
327:
324:
318:
315:
309:
306:
301:
297:
260:
252:
240:
235:
230:
223:
221:
199:
196:
190:
174:
162:
157:
143:
135:
131:
126:
120:
114:
95:
91:
79:
75:
71:
53:
37:
19:
5886:
5684:Ultraproduct
5531:Model theory
5496:Independence
5432:Formal proof
5424:Proof theory
5407:
5380:
5337:real numbers
5309:second-order
5220:Substitution
5097:Metalanguage
5038:conservative
5011:Axiom schema
4955:Constructive
4925:MorseâKelley
4891:Set theories
4870:Aleph number
4863:inaccessible
4769:Grothendieck
4662:
4653:intersection
4540:Higher-order
4528:Second-order
4474:Truth tables
4431:Venn diagram
4214:Formal proof
4086:Georg Cantor
4081:Paul Bernays
4012:MorseâKelley
3987:
3920:
3919:Subset
3866:hereditarily
3828:Venn diagram
3786:ordered pair
3701:Intersection
3679:
3645:Axiom schema
3476:
3457:
3454:"Complement"
3414:
3378:
3356:
3353:Bourbaki, N.
3325:
3320:, p. 6.
3313:
3284:
3275:
3264:. Retrieved
3260:
3237:. Retrieved
3233:
3224:
3108:. A variant
3083:
3049:
2723:
2666:
1901:
1895:
1889:
1886:
1806:real numbers
1558:
1552:
1546:
1540:
1534:
1432:
1426:
1423:
1372:
1366:
1361:
1349:
1343:
1337:
1331:
1326:
1321:
1315:
1310:
1305:
1299:
1296:
1280:
1269:
1265:
1258:
1248:
1245:
1184:
1037:
977:
927:, then
740:
586:
571:
557:proper class
362:
258:
250:
244:
233:
172:
160:
158:
141:
127:
112:
73:
67:
5794:Type theory
5742:undecidable
5674:Truth value
5561:equivalence
5240:non-logical
4853:Enumeration
4843:Isomorphism
4790:cardinality
4774:Von Neumann
4739:Ultrafilter
4704:Uncountable
4638:equivalence
4555:Quantifiers
4545:Fixed-point
4514:First-order
4394:Consistency
4379:Proposition
4356:Traditional
4327:Lindström's
4317:Compactness
4259:Type theory
4204:Cardinality
4111:Thomas Jech
3954:Alternative
3933:Uncountable
3887:Ultrafilter
3746:Cardinality
3650:replacement
3591:Determinacy
3318:Devlin 1979
3306:Halmos 1960
3154:\complement
1436:is denoted
1347:but not in
5909:Categories
5605:elementary
5298:arithmetic
5166:Quantifier
5144:functional
5016:Expression
4734:Transitive
4678:identities
4663:complement
4596:hereditary
4579:Set theory
4106:Kurt Gödel
4091:Paul Cohen
3928:Transitive
3696:Identities
3680:Complement
3667:Operations
3628:Regularity
3596:projective
3559:Adjunction
3518:Set theory
3435:0087.04403
3401:0407.04003
3346:References
3266:2020-09-04
3239:2020-09-04
3167:COMPLEMENT
3102:\backslash
3078:See also:
3064:operations
1911:identities
1877:See also:
1873:Properties
1559:Formally:
1293:Definition
979:Involution
568:Properties
553:set theory
259:complement
241:Definition
183:, written
74:complement
70:set theory
5876:Supertask
5779:Recursion
5737:decidable
5571:saturated
5549:of models
5472:deductive
5467:axiomatic
5387:Hilbert's
5374:Euclidean
5355:canonical
5278:axiomatic
5210:Signature
5139:Predicate
5028:Extension
4950:Ackermann
4875:Operation
4754:Universal
4744:Recursive
4719:Singleton
4714:Inhabited
4699:Countable
4689:Types of
4673:power set
4643:partition
4560:Predicate
4506:Predicate
4421:Syllogism
4411:Soundness
4384:Inference
4374:Tautology
4276:paradoxes
4039:Paradoxes
3959:Axiomatic
3938:Universal
3914:Singleton
3909:Recursive
3852:Countable
3847:Amorphous
3706:Power set
3623:Power set
3574:dependent
3569:countable
3478:MathWorld
3459:MathWorld
3120:∁
3098:\setminus
3094:backslash
3090:\setminus
2867:∖
2858:×
2837:¯
2785:×
2739:¯
2704:×
2641:∖
2635:⊇
2609:∖
2603:⊇
2575:∖
2569:⊃
2563:∖
2537:⊂
2509:∅
2500:∖
2467:∅
2464:∖
2437:∅
2428:∖
2425:∅
2401:∅
2392:∖
2359:∖
2350:∖
2341:∪
2326:∪
2317:∖
2281:∖
2272:∩
2260:∖
2251:∩
2236:∩
2227:∖
2191:∩
2173:∖
2164:∖
2130:∖
2121:∪
2112:∩
2094:∖
2085:∖
2052:∖
2043:∩
2034:∖
2016:∪
2007:∖
1974:∖
1965:∪
1956:∖
1938:∩
1929:∖
1847:∖
1732:∖
1654:∖
1597:∉
1585:∈
1570:∖
1514:−
1477:−
1447:∖
1406:∖
1395:∁
1387:∩
1276:partition
1225:∖
1214:∁
1206:∖
1201:∁
1161:∩
1152:∪
1147:∁
1133:∪
1128:∁
1115:∁
1104:∖
1072:∁
1064:∩
1052:∖
1012:∁
1002:∁
949:∁
941:⊆
936:∁
920:⊆
888:∅
880:∁
843:∁
839:∅
814:∅
806:∁
798:∩
763:∁
755:∪
721:∁
713:∪
708:∁
695:∁
683:∩
648:∁
640:∩
635:∁
622:∁
610:∪
515:congruent
507:multiples
463:∁
443:∁
407:¯
380:∁
340:∉
328:∈
313:∖
302:∁
194:∖
96:∁
18:Set minus
5861:Logicism
5854:timeline
5830:Concrete
5689:Validity
5659:T-schema
5652:Kripke's
5647:Tarski's
5642:semantic
5632:Strength
5581:submodel
5576:spectrum
5544:function
5392:Tarski's
5381:Elements
5368:geometry
5324:Robinson
5245:variable
5230:function
5203:spectrum
5193:Sentence
5149:variable
5092:Language
5045:Relation
5006:Automata
4996:Alphabet
4980:language
4834:-jection
4812:codomain
4798:Function
4759:Universe
4729:Infinite
4633:Relation
4416:Validity
4406:Argument
4304:theorem,
4043:Problems
3947:Theories
3923:Superset
3899:Infinite
3728:Concepts
3608:Infinity
3525:Overview
3411:(1960).
3383:Springer
3377:(1979).
3355:(1970).
3332:Archived
3173:See also
3164:∁
1621:Examples
913:If
491:integers
484:Examples
419:′
130:universe
119:elements
5803:Related
5600:Diagram
5498: (
5477:Hilbert
5462:Systems
5457:Theorem
5335:of the
5280:systems
5060:Formula
5055:Grammar
4971: (
4915:General
4628:Forcing
4613:Element
4533:Monadic
4308:paradox
4249:Theorem
4185:General
3981:General
3976:Zermelo
3882:subbase
3864: (
3803:Forcing
3781:Element
3753: (
3731:Methods
3618:Pairing
3084:In the
3066:of the
3004:column
2552:, then
1834:, then
1262:, then
532:is the
134:members
121:not in
5566:finite
5329:Skolem
5282:
5257:Theory
5225:Symbol
5215:String
5198:atomic
5075:ground
5070:closed
5065:atomic
5021:ground
4984:syntax
4880:binary
4807:domain
4724:Finite
4489:finite
4347:Logics
4306:
4254:Theory
3872:Filter
3862:Finite
3798:Family
3741:Almost
3579:global
3564:Choice
3551:Axioms
3433:
3423:
3399:
3389:
3363:
3161:
3159:U+2201
2885:Here,
2849:
2843:
1899:, and
1532:where
140:, the
72:, the
5556:Model
5304:Peano
5161:Proof
5001:Arity
4930:Naive
4817:image
4749:Fuzzy
4709:Empty
4658:union
4603:Class
4244:Model
4234:Lemma
4192:Axiom
3964:Naive
3894:Fuzzy
3857:Empty
3840:types
3791:tuple
3761:Class
3755:large
3716:Union
3633:Union
3216:Notes
3086:LaTeX
1550:from
1274:is a
534:union
493:. If
76:of a
5679:Type
5482:list
5286:list
5263:list
5252:Term
5186:rank
5080:open
4974:list
4786:Maps
4691:sets
4550:Free
4520:list
4270:list
4197:list
3877:base
3421:ISBN
3387:ISBN
3361:ISBN
3054:and
2722:The
1887:Let
1881:and
1808:and
1544:and
1360:The
1335:and
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576:and
572:Let
232:The
179:and
159:The
110:(or
5366:of
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3169:.)
2774:in
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3296:^
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2019:B
2013:A
2010:(
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1980:)
1977:B
1971:C
1968:(
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1959:A
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1950:(
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