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Sets cannot have duplicate elements, so the union of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}. Multiple occurrences of identical elements have no effect on the
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1937:"Set Operations | Union | Intersection | Complement | Difference | Mutually Exclusive | Partitions | De Morgan's Law | Distributive Law | Cartesian Product"
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in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A
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When the symbol "âȘ" is placed before other symbols (instead of between them), it is usually rendered as a larger size.
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891:. In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formula
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Thus, the parentheses may be omitted without ambiguity: either of the above can be written as
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The most general notion is the union of an arbitrary collection of sets, sometimes called an
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One can take the union of several sets simultaneously. For example, the union of three sets
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is the union of a finite number of sets; the phrase does not imply that the union set is a
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The notation for the general concept can vary considerably. For a finite union of sets
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304:= {1, 2, 3, 4, 5, 6, 7}. A more elaborate example (involving two infinite sets) is:
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1202:{\displaystyle x\in \bigcup \mathbf {M} \iff \exists A\in \mathbf {M} ,\ x\in A.}
434:
2134:"The Unicode Standard, Version 15.0 â Mathematical Operators â Range: 2200â22FF"
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952:{\displaystyle A\cup B=(A^{\complement }\cap B^{\complement })^{\complement },}
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1550:. The last of these notations refers to the union of the collection
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The union of A, B, C, D, and E is everything except the white area.
177:
For explanation of the symbols used in this article, refer to the
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67:
2104:
Smith, Douglas; Eggen, Maurice; Andre, Richard St (2014-08-01).
2607:
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De Morgan's laws formally proven from the axioms of set theory.
1389:{\displaystyle S_{1}\cup S_{2}\cup S_{3}\cup \dots \cup S_{n}}
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414:{2, 4, 6, 8, 10, ...}, because 9 is neither prime nor even.
143:
30:
1872: â Repeated application of an operation to a sequence
693:. All these properties follow from analogous facts about
1753:
1444:. Various common notations for arbitrary unions include
277:{\displaystyle A\cup B=\{x:x\in A{\text{ or }}x\in B\}}
844:{\displaystyle A\cup (B\cap C)=(A\cup B)\cap (A\cup C).}
1212:
This idea subsumes the preceding sectionsâfor example,
767:{\displaystyle A\cap (B\cup C)=(A\cap B)\cup (A\cap C)}
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Pages displaying wikidata descriptions as a fallback
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1832: â Identities and relationships involving sets
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1866: â Set of elements common to all of some sets
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1989:
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542:{\displaystyle A\cup (B\cup C)=(A\cup B)\cup C.}
2103:
879:, together with the operations given by union,
589:, so the sets can be written in any order. The
1993:Applied Mathematics for Database Professionals
1317:{\displaystyle S_{1},S_{2},S_{3},\dots ,S_{n}}
2851:
2198:
2177:Infinite Union and Intersection at ProvenMath
1990:deHaan, Lex; Koppelaars, Toon (2007-10-25).
392:{\displaystyle A\cup B=\{2,3,4,5,6,\dots \}}
386:
350:
271:
239:
1890: â Elements in exactly one of two sets
1878: â Equalities for combinations of sets
1860: â Counting technique in combinatorics
1592:{\displaystyle \left\{A_{i}:i\in I\right\}}
1240:is the empty collection, then the union of
657:. Also, the union operation is idempotent:
3043:
2858:
2844:
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2191:
1713:{\textstyle \bigcup _{i=1}^{\infty }A_{i}}
1500:{\textstyle \bigcup _{A\in \mathbf {M} }A}
1166:
1162:
170:sets and it is by definition equal to the
2050:. Springer Science & Business Media.
2023:. Springer Science & Business Media.
1854: â In mathematics, operation on sets
2043:
1739:, union is represented by the character
774:and union distributes over intersection
109:
66:
29:
2072:"Finite Union of Finite Sets is Finite"
1848: â Concept in axiomatic set theory
424:
14:
4596:
2865:
2082:from the original on 11 September 2014
2016:
2839:
2186:
1906:
597:for the operation of union. That is,
410:{2, 3, 5, 7, 11, ...} and the set of
406:contained in the union of the set of
2108:A Transition to Advanced Mathematics
2099:
2097:
1958:
1956:
1931:
1929:
1876:List of set identities and relations
1836:Alternation (formal language theory)
1730:
1720:, which is analogous to that of the
1437:{\textstyle \bigcup _{i=1}^{n}S_{i}}
700:Intersection distributes over union
624:{\displaystyle A\cup \varnothing =A}
431:List of set identities and relations
402:As another example, the number 9 is
197:is the set of elements which are in
184:
1543:{\textstyle \bigcup _{i\in I}A_{i}}
1090:
475:{\displaystyle A,B,{\text{ and }}C}
27:Set of elements in any of some sets
24:
1695:
1167:
126:(denoted by âȘ) of a collection of
25:
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2152:
2094:
1953:
1926:
1664:. In the case that the index set
1462:{\textstyle \bigcup \mathbf {M} }
978:{\displaystyle {}^{\complement }}
612:
443:operation; that is, for any sets
4577:
2236:
2044:Dasgupta, Abhijit (2013-12-11).
1814:
1488:
1455:
1224:is the union of the collection {
1177:
1158:
1015:
329:is an odd integer larger than 1}
1915:from the original on 2009-02-07
2212:
2126:
2064:
2037:
2010:
1983:
1900:
1842:â the union of sets of strings
1163:
1111:is an element of the union of
1107:whose elements are sets, then
985:denotes the complement in the
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497:
100:{\displaystyle ~A\cup B\cup C}
13:
1:
4538:History of mathematical logic
1969:. American Mathematical Soc.
1884: â Informal set theories
1858:Inclusionâexclusion principle
576:{\displaystyle A\cup B\cup C}
179:table of mathematical symbols
4604:Basic concepts in set theory
4463:Primitive recursive function
2017:Halmos, P. R. (2013-11-27).
1247:
7:
2165:Encyclopedia of Mathematics
1807:
10:
4635:
3527:SchröderâBernstein theorem
3254:Monadic predicate calculus
2913:Foundations of mathematics
2696:von NeumannâBernaysâGödel
1044:, and nothing else. Thus,
428:
421:of a set or its contents.
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4560:Philosophy of mathematics
4509:Automated theorem proving
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2497:One-to-one correspondence
2413:
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1870:Iterated binary operation
1864:Intersection (set theory)
1032:contains all elements of
684:{\displaystyle A\cup A=A}
1894:
1672:, one uses the notation
57:{\displaystyle ~A\cup B}
4210:Self-verifying theories
4031:Tarski's axiomatization
2982:Tarski's undefinability
2977:incompleteness theorems
296:= {1, 2, 4, 6, 7} then
4584:Mathematics portal
4195:Proof of impossibility
3843:propositional variable
3153:Propositional calculus
2455:Constructible universe
2282:Constructibility (V=L)
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1655:{\displaystyle i\in I}
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1064:is in at least one of
1040:, and all elements of
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959:where the superscript
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189:The union of two sets
160:
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101:
64:
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4453:Kolmogorov complexity
4406:Computably enumerable
4306:Model complete theory
4098:Principia Mathematica
3158:Propositional formula
2987:BanachâTarski paradox
2678:Principia Mathematica
2512:Transfinite induction
2371:(i.e. set difference)
1911:. Wolfram Mathworld.
1795:
1793:{\textstyle \bigcup }
1771:
1769:{\displaystyle \cup }
1715:
1679:
1657:
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1627:{\displaystyle A_{i}}
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279:
161:
142:refers to a union of
113:
102:
70:
59:
33:
4401:ChurchâTuring thesis
4388:Computability theory
3597:continuum hypothesis
3115:Square of opposition
2973:Gödel's completeness
2752:Burali-Forti paradox
2507:Set-builder notation
2460:Continuum hypothesis
2400:Symmetric difference
2112:. Cengage Learning.
1888:Symmetric difference
1784:
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425:Algebraic properties
335:
224:
215:set-builder notation
150:
76:
71:Union of three sets:
39:
4555:Mathematical object
4446:P versus NP problem
4411:Computable function
4205:Reverse mathematics
4131:Logical consequence
4008:primitive recursive
4003:elementary function
3776:Free/bound variable
3629:TarskiâGrothendieck
3148:Logical connectives
3078:Logical equivalence
2928:Logical consequence
2713:TarskiâGrothendieck
1907:Weisstein, Eric W.
1634:is a set for every
695:logical disjunction
439:Binary union is an
292:= {1, 3, 5, 7} and
4614:Operations on sets
4353:Transfer principle
4316:Semantics of logic
4301:Categorical theory
4277:Non-standard model
3791:Logical connective
2918:Information theory
2867:Mathematical logic
2302:Limitation of size
1941:Probability Course
1822:Mathematics portal
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1244:is the empty set.
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130:is the set of all
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34:Union of two sets:
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4523:Abstract category
4326:Theories of truth
4136:Rule of inference
4126:Natural deduction
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3357:Cartesian product
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3261:
3168:Many-valued logic
3143:Boolean functions
3026:Russell's paradox
3001:diagonal argument
2898:First-order logic
2833:
2832:
2742:Russell's paradox
2691:ZermeloâFraenkel
2592:Dedekind-infinite
2465:Diagonal argument
2364:Cartesian product
2228:Set (mathematics)
1800:is rendered from
1776:is rendered from
1731:Notation encoding
1514:
1476:
1324:one often writes
1186:
1134:is an element of
1048:is an element of
1003:{\displaystyle U}
870:{\displaystyle U}
648:{\displaystyle A}
585:. Also, union is
467:
260:
185:Union of two sets
159:{\displaystyle 0}
81:
44:
16:(Redirected from
4626:
4582:
4581:
4533:History of logic
4528:Category of sets
4421:Decision problem
4200:Ordinal analysis
4141:Sequent calculus
4039:Boolean algebras
3979:
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3924:logical/constant
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3587:ZermeloâFraenkel
3338:Set operations:
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3021:LöwenheimâSkolem
2908:Formal semantics
2860:
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2815:Bertrand Russell
2805:John von Neumann
2790:Abraham Fraenkel
2785:Richard Dedekind
2747:Suslin's problem
2658:Cantor's theorem
2375:De Morgan's laws
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1966:Basic Set Theory
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1097:infinitary union
1091:Arbitrary unions
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656:
654:
652:
651:
646:
632:
630:
628:
627:
622:
595:identity element
584:
582:
580:
579:
574:
548:
546:
545:
540:
483:
481:
479:
478:
473:
468:
465:
398:
396:
395:
390:
288:For example, if
283:
281:
280:
275:
261:
258:
167:
165:
163:
162:
157:
140:
139:
106:
104:
103:
98:
79:
63:
61:
60:
55:
42:
21:
4634:
4633:
4629:
4628:
4627:
4625:
4624:
4623:
4609:Boolean algebra
4594:
4593:
4592:
4587:
4576:
4569:
4514:Category theory
4504:Algebraic logic
4487:
4458:Lambda calculus
4396:Church encoding
4382:
4358:Truth predicate
4214:
4180:Complete theory
4103:
3972:
3968:
3964:
3959:
3951:
3671: and
3667:
3662:
3648:
3624:New Foundations
3592:axiom of choice
3575:
3537:Gödel numbering
3477: and
3469:
3373:
3258:
3208:
3189:
3138:Boolean algebra
3124:
3088:Equiconsistency
3053:Classical logic
3030:
3011:Halting problem
2999: and
2975: and
2963: and
2962:
2957:Theorems (
2952:
2869:
2864:
2834:
2829:
2756:
2735:
2719:
2684:New Foundations
2631:
2521:
2440:Cardinal number
2423:
2409:
2350:
2241:
2232:
2216:
2211:
2160:"Union of sets"
2158:
2155:
2150:
2136:
2132:
2131:
2127:
2120:
2102:
2095:
2085:
2083:
2070:
2069:
2065:
2058:
2042:
2038:
2031:
2015:
2011:
2004:
1988:
1984:
1977:
1961:
1954:
1945:
1943:
1935:
1934:
1927:
1918:
1916:
1905:
1901:
1897:
1839:
1830:Algebra of sets
1820:
1813:
1810:
1801:
1785:
1782:
1781:
1777:
1761:
1758:
1757:
1749:
1746:
1741:
1740:
1733:
1704:
1700:
1694:
1683:
1677:
1674:
1673:
1670:natural numbers
1641:
1638:
1637:
1635:
1618:
1614:
1612:
1609:
1608:
1566:
1562:
1561:
1557:
1555:
1552:
1551:
1534:
1530:
1518:
1512:
1509:
1508:
1487:
1480:
1474:
1471:
1470:
1454:
1449:
1446:
1445:
1428:
1424:
1418:
1407:
1401:
1398:
1397:
1380:
1376:
1361:
1357:
1348:
1344:
1335:
1331:
1329:
1326:
1325:
1308:
1304:
1289:
1285:
1276:
1272:
1263:
1259:
1257:
1254:
1253:
1250:
1176:
1157:
1146:
1143:
1142:
1093:
1060:if and only if
1018:
995:
992:
991:
989:
969:
967:
966:
964:
961:
960:
940:
936:
930:
926:
917:
913:
896:
893:
892:
889:Boolean algebra
885:complementation
862:
859:
858:
856:
779:
776:
775:
705:
702:
701:
664:
661:
660:
658:
640:
637:
636:
634:
604:
601:
600:
598:
556:
553:
552:
550:
489:
486:
485:
466: and
464:
450:
447:
446:
444:
437:
435:Algebra of sets
427:
336:
333:
332:
257:
225:
222:
221:
187:
151:
148:
147:
145:
137:
136:
77:
74:
73:
72:
40:
37:
36:
35:
28:
23:
22:
15:
12:
11:
5:
4632:
4622:
4621:
4616:
4611:
4606:
4589:
4588:
4574:
4571:
4570:
4568:
4567:
4562:
4557:
4552:
4547:
4546:
4545:
4535:
4530:
4525:
4516:
4511:
4506:
4501:
4499:Abstract logic
4495:
4493:
4489:
4488:
4486:
4485:
4480:
4478:Turing machine
4475:
4470:
4465:
4460:
4455:
4450:
4449:
4448:
4443:
4438:
4433:
4428:
4418:
4416:Computable set
4413:
4408:
4403:
4398:
4392:
4390:
4384:
4383:
4381:
4380:
4375:
4370:
4365:
4360:
4355:
4350:
4345:
4344:
4343:
4338:
4333:
4323:
4318:
4313:
4311:Satisfiability
4308:
4303:
4298:
4297:
4296:
4286:
4285:
4284:
4274:
4273:
4272:
4267:
4262:
4257:
4252:
4242:
4241:
4240:
4235:
4228:Interpretation
4224:
4222:
4216:
4215:
4213:
4212:
4207:
4202:
4197:
4192:
4182:
4177:
4176:
4175:
4174:
4173:
4163:
4158:
4148:
4143:
4138:
4133:
4128:
4123:
4117:
4115:
4109:
4108:
4105:
4104:
4102:
4101:
4093:
4092:
4091:
4090:
4085:
4084:
4083:
4078:
4073:
4053:
4052:
4051:
4049:minimal axioms
4046:
4035:
4034:
4033:
4022:
4021:
4020:
4015:
4010:
4005:
4000:
3995:
3982:
3980:
3961:
3960:
3958:
3957:
3956:
3955:
3943:
3938:
3937:
3936:
3931:
3926:
3921:
3911:
3906:
3901:
3896:
3895:
3894:
3889:
3879:
3878:
3877:
3872:
3867:
3862:
3852:
3847:
3846:
3845:
3840:
3835:
3825:
3824:
3823:
3818:
3813:
3808:
3803:
3798:
3788:
3783:
3778:
3773:
3772:
3771:
3766:
3761:
3756:
3746:
3741:
3739:Formation rule
3736:
3731:
3730:
3729:
3724:
3714:
3713:
3712:
3702:
3697:
3692:
3687:
3681:
3675:
3658:Formal systems
3654:
3653:
3650:
3649:
3647:
3646:
3641:
3636:
3631:
3626:
3621:
3616:
3611:
3606:
3601:
3600:
3599:
3594:
3583:
3581:
3577:
3576:
3574:
3573:
3572:
3571:
3561:
3556:
3555:
3554:
3547:Large cardinal
3544:
3539:
3534:
3529:
3524:
3510:
3509:
3508:
3503:
3498:
3483:
3481:
3471:
3470:
3468:
3467:
3466:
3465:
3460:
3455:
3445:
3440:
3435:
3430:
3425:
3420:
3415:
3410:
3405:
3400:
3395:
3390:
3384:
3382:
3375:
3374:
3372:
3371:
3370:
3369:
3364:
3359:
3354:
3349:
3344:
3336:
3335:
3334:
3329:
3319:
3314:
3312:Extensionality
3309:
3307:Ordinal number
3304:
3294:
3289:
3288:
3287:
3276:
3270:
3264:
3263:
3260:
3259:
3257:
3256:
3251:
3246:
3241:
3236:
3231:
3226:
3225:
3224:
3214:
3213:
3212:
3199:
3197:
3191:
3190:
3188:
3187:
3186:
3185:
3180:
3175:
3165:
3160:
3155:
3150:
3145:
3140:
3134:
3132:
3126:
3125:
3123:
3122:
3117:
3112:
3107:
3102:
3097:
3092:
3091:
3090:
3080:
3075:
3070:
3065:
3060:
3055:
3049:
3047:
3038:
3032:
3031:
3029:
3028:
3023:
3018:
3013:
3008:
3003:
2991:Cantor's
2989:
2984:
2979:
2969:
2967:
2954:
2953:
2951:
2950:
2945:
2940:
2935:
2930:
2925:
2920:
2915:
2910:
2905:
2900:
2895:
2890:
2889:
2888:
2877:
2875:
2871:
2870:
2863:
2862:
2855:
2848:
2840:
2831:
2830:
2828:
2827:
2822:
2820:Thoralf Skolem
2817:
2812:
2807:
2802:
2797:
2792:
2787:
2782:
2777:
2772:
2766:
2764:
2758:
2757:
2755:
2754:
2749:
2744:
2738:
2736:
2734:
2733:
2730:
2724:
2721:
2720:
2718:
2717:
2716:
2715:
2710:
2705:
2704:
2703:
2688:
2687:
2686:
2674:
2673:
2672:
2661:
2660:
2655:
2650:
2645:
2639:
2637:
2633:
2632:
2630:
2629:
2624:
2619:
2614:
2605:
2600:
2595:
2585:
2580:
2579:
2578:
2573:
2568:
2558:
2548:
2543:
2538:
2532:
2530:
2523:
2522:
2520:
2519:
2514:
2509:
2504:
2502:Ordinal number
2499:
2494:
2489:
2484:
2483:
2482:
2477:
2467:
2462:
2457:
2452:
2447:
2437:
2432:
2426:
2424:
2422:
2421:
2418:
2414:
2411:
2410:
2408:
2407:
2402:
2397:
2392:
2387:
2382:
2380:Disjoint union
2377:
2372:
2366:
2360:
2358:
2352:
2351:
2349:
2348:
2347:
2346:
2341:
2330:
2329:
2327:Martin's axiom
2324:
2319:
2314:
2309:
2304:
2299:
2294:
2292:Extensionality
2289:
2284:
2279:
2278:
2277:
2272:
2267:
2257:
2251:
2249:
2243:
2242:
2235:
2233:
2231:
2230:
2224:
2222:
2218:
2217:
2210:
2209:
2202:
2195:
2187:
2181:
2180:
2174:
2154:
2153:External links
2151:
2149:
2148:
2125:
2118:
2093:
2063:
2056:
2036:
2029:
2009:
2002:
1982:
1975:
1952:
1925:
1898:
1896:
1893:
1892:
1891:
1885:
1879:
1873:
1867:
1861:
1855:
1852:Disjoint union
1849:
1846:Axiom of union
1843:
1833:
1826:
1825:
1809:
1806:
1789:
1765:
1732:
1729:
1707:
1703:
1697:
1692:
1689:
1686:
1682:
1668:is the set of
1651:
1648:
1645:
1621:
1617:
1587:
1583:
1580:
1577:
1574:
1569:
1565:
1560:
1537:
1533:
1527:
1524:
1521:
1517:
1496:
1490:
1486:
1483:
1479:
1457:
1453:
1431:
1427:
1421:
1416:
1413:
1410:
1406:
1383:
1379:
1375:
1372:
1369:
1364:
1360:
1356:
1351:
1347:
1343:
1338:
1334:
1311:
1307:
1303:
1300:
1297:
1292:
1288:
1284:
1279:
1275:
1271:
1266:
1262:
1249:
1246:
1210:
1209:
1198:
1195:
1192:
1189:
1183:
1179:
1175:
1172:
1169:
1165:
1160:
1156:
1153:
1150:
1138:. In symbols:
1116:if and only if
1092:
1089:
1017:
1014:
999:
972:
948:
943:
939:
933:
929:
925:
920:
916:
912:
909:
906:
903:
900:
866:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
798:
795:
792:
789:
786:
783:
763:
760:
757:
754:
751:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
680:
677:
674:
671:
668:
644:
633:, for any set
620:
617:
614:
611:
608:
572:
569:
566:
563:
560:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
508:
505:
502:
499:
496:
493:
471:
463:
460:
457:
454:
426:
423:
400:
399:
388:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
330:
320:
319:larger than 1}
286:
285:
273:
270:
267:
264:
259: or
256:
253:
250:
247:
244:
241:
238:
235:
232:
229:
186:
183:
155:
96:
93:
90:
87:
84:
53:
50:
47:
26:
9:
6:
4:
3:
2:
4631:
4620:
4617:
4615:
4612:
4610:
4607:
4605:
4602:
4601:
4599:
4586:
4585:
4580:
4572:
4566:
4563:
4561:
4558:
4556:
4553:
4551:
4548:
4544:
4541:
4540:
4539:
4536:
4534:
4531:
4529:
4526:
4524:
4520:
4517:
4515:
4512:
4510:
4507:
4505:
4502:
4500:
4497:
4496:
4494:
4490:
4484:
4481:
4479:
4476:
4474:
4473:Recursive set
4471:
4469:
4466:
4464:
4461:
4459:
4456:
4454:
4451:
4447:
4444:
4442:
4439:
4437:
4434:
4432:
4429:
4427:
4424:
4423:
4422:
4419:
4417:
4414:
4412:
4409:
4407:
4404:
4402:
4399:
4397:
4394:
4393:
4391:
4389:
4385:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4351:
4349:
4346:
4342:
4339:
4337:
4334:
4332:
4329:
4328:
4327:
4324:
4322:
4319:
4317:
4314:
4312:
4309:
4307:
4304:
4302:
4299:
4295:
4292:
4291:
4290:
4287:
4283:
4282:of arithmetic
4280:
4279:
4278:
4275:
4271:
4268:
4266:
4263:
4261:
4258:
4256:
4253:
4251:
4248:
4247:
4246:
4243:
4239:
4236:
4234:
4231:
4230:
4229:
4226:
4225:
4223:
4221:
4217:
4211:
4208:
4206:
4203:
4201:
4198:
4196:
4193:
4190:
4189:from ZFC
4186:
4183:
4181:
4178:
4172:
4169:
4168:
4167:
4164:
4162:
4159:
4157:
4154:
4153:
4152:
4149:
4147:
4144:
4142:
4139:
4137:
4134:
4132:
4129:
4127:
4124:
4122:
4119:
4118:
4116:
4114:
4110:
4100:
4099:
4095:
4094:
4089:
4088:non-Euclidean
4086:
4082:
4079:
4077:
4074:
4072:
4071:
4067:
4066:
4064:
4061:
4060:
4058:
4054:
4050:
4047:
4045:
4042:
4041:
4040:
4036:
4032:
4029:
4028:
4027:
4023:
4019:
4016:
4014:
4011:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3990:
3988:
3984:
3983:
3981:
3976:
3970:
3965:Example
3962:
3954:
3949:
3948:
3947:
3944:
3942:
3939:
3935:
3932:
3930:
3927:
3925:
3922:
3920:
3917:
3916:
3915:
3912:
3910:
3907:
3905:
3902:
3900:
3897:
3893:
3890:
3888:
3885:
3884:
3883:
3880:
3876:
3873:
3871:
3868:
3866:
3863:
3861:
3858:
3857:
3856:
3853:
3851:
3848:
3844:
3841:
3839:
3836:
3834:
3831:
3830:
3829:
3826:
3822:
3819:
3817:
3814:
3812:
3809:
3807:
3804:
3802:
3799:
3797:
3794:
3793:
3792:
3789:
3787:
3784:
3782:
3779:
3777:
3774:
3770:
3767:
3765:
3762:
3760:
3757:
3755:
3752:
3751:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3728:
3725:
3723:
3722:by definition
3720:
3719:
3718:
3715:
3711:
3708:
3707:
3706:
3703:
3701:
3698:
3696:
3693:
3691:
3688:
3686:
3683:
3682:
3679:
3676:
3674:
3670:
3665:
3659:
3655:
3645:
3642:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3615:
3612:
3610:
3609:KripkeâPlatek
3607:
3605:
3602:
3598:
3595:
3593:
3590:
3589:
3588:
3585:
3584:
3582:
3578:
3570:
3567:
3566:
3565:
3562:
3560:
3557:
3553:
3550:
3549:
3548:
3545:
3543:
3540:
3538:
3535:
3533:
3530:
3528:
3525:
3522:
3518:
3514:
3511:
3507:
3504:
3502:
3499:
3497:
3494:
3493:
3492:
3488:
3485:
3484:
3482:
3480:
3476:
3472:
3464:
3461:
3459:
3456:
3454:
3453:constructible
3451:
3450:
3449:
3446:
3444:
3441:
3439:
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3385:
3383:
3381:
3376:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3339:
3337:
3333:
3330:
3328:
3325:
3324:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3299:
3295:
3293:
3290:
3286:
3283:
3282:
3281:
3278:
3277:
3274:
3271:
3269:
3265:
3255:
3252:
3250:
3247:
3245:
3242:
3240:
3237:
3235:
3232:
3230:
3227:
3223:
3220:
3219:
3218:
3215:
3211:
3206:
3205:
3204:
3201:
3200:
3198:
3196:
3192:
3184:
3181:
3179:
3176:
3174:
3171:
3170:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3136:
3135:
3133:
3131:
3130:Propositional
3127:
3121:
3118:
3116:
3113:
3111:
3108:
3106:
3103:
3101:
3098:
3096:
3093:
3089:
3086:
3085:
3084:
3081:
3079:
3076:
3074:
3071:
3069:
3066:
3064:
3061:
3059:
3058:Logical truth
3056:
3054:
3051:
3050:
3048:
3046:
3042:
3039:
3037:
3033:
3027:
3024:
3022:
3019:
3017:
3014:
3012:
3009:
3007:
3004:
3002:
2998:
2994:
2990:
2988:
2985:
2983:
2980:
2978:
2974:
2971:
2970:
2968:
2966:
2960:
2955:
2949:
2946:
2944:
2941:
2939:
2936:
2934:
2931:
2929:
2926:
2924:
2921:
2919:
2916:
2914:
2911:
2909:
2906:
2904:
2901:
2899:
2896:
2894:
2891:
2887:
2884:
2883:
2882:
2879:
2878:
2876:
2872:
2868:
2861:
2856:
2854:
2849:
2847:
2842:
2841:
2838:
2826:
2825:Ernst Zermelo
2823:
2821:
2818:
2816:
2813:
2811:
2810:Willard Quine
2808:
2806:
2803:
2801:
2798:
2796:
2793:
2791:
2788:
2786:
2783:
2781:
2778:
2776:
2773:
2771:
2768:
2767:
2765:
2763:
2762:Set theorists
2759:
2753:
2750:
2748:
2745:
2743:
2740:
2739:
2737:
2731:
2729:
2726:
2725:
2722:
2714:
2711:
2709:
2708:KripkeâPlatek
2706:
2702:
2699:
2698:
2697:
2694:
2693:
2692:
2689:
2685:
2682:
2681:
2680:
2679:
2675:
2671:
2668:
2667:
2666:
2663:
2662:
2659:
2656:
2654:
2651:
2649:
2646:
2644:
2641:
2640:
2638:
2634:
2628:
2625:
2623:
2620:
2618:
2615:
2613:
2611:
2606:
2604:
2601:
2599:
2596:
2593:
2589:
2586:
2584:
2581:
2577:
2574:
2572:
2569:
2567:
2564:
2563:
2562:
2559:
2556:
2552:
2549:
2547:
2544:
2542:
2539:
2537:
2534:
2533:
2531:
2528:
2524:
2518:
2515:
2513:
2510:
2508:
2505:
2503:
2500:
2498:
2495:
2493:
2490:
2488:
2485:
2481:
2478:
2476:
2473:
2472:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2445:
2441:
2438:
2436:
2433:
2431:
2428:
2427:
2425:
2419:
2416:
2415:
2412:
2406:
2403:
2401:
2398:
2396:
2393:
2391:
2388:
2386:
2383:
2381:
2378:
2376:
2373:
2370:
2367:
2365:
2362:
2361:
2359:
2357:
2353:
2345:
2344:specification
2342:
2340:
2337:
2336:
2335:
2332:
2331:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2308:
2305:
2303:
2300:
2298:
2295:
2293:
2290:
2288:
2285:
2283:
2280:
2276:
2273:
2271:
2268:
2266:
2263:
2262:
2261:
2258:
2256:
2253:
2252:
2250:
2248:
2244:
2239:
2229:
2226:
2225:
2223:
2219:
2215:
2208:
2203:
2201:
2196:
2194:
2189:
2188:
2185:
2178:
2175:
2171:
2167:
2166:
2161:
2157:
2156:
2144:
2143:
2135:
2129:
2121:
2119:9781285463261
2115:
2110:
2109:
2100:
2098:
2081:
2077:
2073:
2067:
2059:
2057:9781461488545
2053:
2049:
2048:
2040:
2032:
2030:9781475716450
2026:
2022:
2021:
2013:
2005:
2003:9781430203483
1999:
1995:
1994:
1986:
1978:
1976:9780821827314
1972:
1968:
1967:
1959:
1957:
1942:
1938:
1932:
1930:
1914:
1910:
1903:
1899:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1837:
1834:
1831:
1828:
1827:
1823:
1817:
1812:
1805:
1787:
1763:
1755:
1738:
1728:
1725:
1723:
1722:infinite sums
1705:
1701:
1690:
1687:
1684:
1680:
1671:
1667:
1649:
1646:
1643:
1619:
1615:
1606:
1602:
1585:
1581:
1578:
1575:
1572:
1567:
1563:
1558:
1535:
1531:
1525:
1522:
1519:
1515:
1494:
1484:
1481:
1477:
1451:
1429:
1425:
1419:
1414:
1411:
1408:
1404:
1381:
1377:
1373:
1370:
1367:
1362:
1358:
1354:
1349:
1345:
1341:
1336:
1332:
1309:
1305:
1301:
1298:
1295:
1290:
1286:
1282:
1277:
1273:
1269:
1264:
1260:
1245:
1243:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1196:
1193:
1190:
1187:
1181:
1173:
1170:
1154:
1151:
1148:
1141:
1140:
1139:
1137:
1133:
1129:
1125:
1121:
1117:
1114:
1110:
1106:
1102:
1098:
1088:
1086:
1082:
1077:
1075:
1071:
1067:
1063:
1059:
1055:
1051:
1047:
1043:
1039:
1035:
1031:
1027:
1023:
1016:Finite unions
1013:
997:
988:
987:universal set
970:
946:
941:
931:
927:
923:
918:
914:
907:
904:
901:
898:
890:
886:
882:
864:
854:
838:
832:
829:
826:
820:
814:
811:
808:
802:
796:
793:
790:
784:
781:
758:
755:
752:
746:
740:
737:
734:
728:
722:
719:
716:
710:
707:
698:
696:
678:
675:
672:
669:
666:
642:
618:
615:
609:
606:
596:
592:
588:
570:
567:
564:
561:
558:
536:
533:
530:
524:
521:
518:
512:
506:
503:
500:
494:
491:
469:
461:
458:
455:
452:
442:
436:
432:
422:
420:
415:
413:
409:
408:prime numbers
405:
383:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
347:
344:
341:
338:
331:
328:
324:
321:
318:
314:
310:
307:
306:
305:
303:
299:
295:
291:
268:
265:
262:
254:
251:
248:
245:
242:
236:
233:
230:
227:
220:
219:
218:
216:
212:
208:
205:, or in both
204:
200:
196:
192:
182:
180:
175:
173:
169:
153:
141:
138:nullary union
133:
129:
125:
121:
112:
94:
91:
88:
85:
82:
69:
51:
48:
45:
32:
19:
4575:
4373:Ultraproduct
4220:Model theory
4185:Independence
4121:Formal proof
4113:Proof theory
4096:
4069:
4026:real numbers
3998:second-order
3909:Substitution
3786:Metalanguage
3727:conservative
3700:Axiom schema
3644:Constructive
3614:MorseâKelley
3580:Set theories
3559:Aleph number
3552:inaccessible
3458:Grothendieck
3346:
3342:intersection
3229:Higher-order
3217:Second-order
3163:Truth tables
3120:Venn diagram
2903:Formal proof
2775:Georg Cantor
2770:Paul Bernays
2701:MorseâKelley
2676:
2609:
2608:Subset
2555:hereditarily
2517:Venn diagram
2475:ordered pair
2404:
2390:Intersection
2334:Axiom schema
2163:
2145:. p. 3.
2140:
2128:
2107:
2084:. Retrieved
2075:
2066:
2046:
2039:
2019:
2012:
1992:
1985:
1965:
1944:. Retrieved
1940:
1917:. Retrieved
1902:
1734:
1726:
1665:
1600:
1251:
1241:
1237:
1236:}. Also, if
1233:
1229:
1225:
1221:
1217:
1213:
1211:
1135:
1131:
1127:
1123:
1120:at least one
1112:
1108:
1103:is a set or
1100:
1096:
1094:
1081:finite union
1080:
1078:
1073:
1069:
1065:
1061:
1057:
1053:
1049:
1045:
1041:
1037:
1033:
1029:
1025:
1021:
1019:
881:intersection
699:
438:
416:
412:even numbers
403:
401:
326:
322:
312:
308:
301:
297:
293:
289:
287:
210:
206:
202:
198:
194:
190:
188:
176:
135:
123:
117:
4483:Type theory
4431:undecidable
4363:Truth value
4250:equivalence
3929:non-logical
3542:Enumeration
3532:Isomorphism
3479:cardinality
3463:Von Neumann
3428:Ultrafilter
3393:Uncountable
3327:equivalence
3244:Quantifiers
3234:Fixed-point
3203:First-order
3083:Consistency
3068:Proposition
3045:Traditional
3016:Lindström's
3006:Compactness
2948:Type theory
2893:Cardinality
2800:Thomas Jech
2643:Alternative
2622:Uncountable
2576:Ultrafilter
2435:Cardinality
2339:replacement
2287:Determinacy
1724:in series.
587:commutative
441:associative
419:cardinality
315:is an even
4619:Set theory
4598:Categories
4294:elementary
3987:arithmetic
3855:Quantifier
3833:functional
3705:Expression
3423:Transitive
3367:identities
3352:complement
3285:hereditary
3268:Set theory
2795:Kurt Gödel
2780:Paul Cohen
2617:Transitive
2385:Identities
2369:Complement
2356:Operations
2317:Regularity
2255:Adjunction
2214:Set theory
1996:. Apress.
1946:2020-09-05
1919:2009-07-14
1130:such that
1085:finite set
429:See also:
120:set theory
4565:Supertask
4468:Recursion
4426:decidable
4260:saturated
4238:of models
4161:deductive
4156:axiomatic
4076:Hilbert's
4063:Euclidean
4044:canonical
3967:axiomatic
3899:Signature
3828:Predicate
3717:Extension
3639:Ackermann
3564:Operation
3443:Universal
3433:Recursive
3408:Singleton
3403:Inhabited
3388:Countable
3378:Types of
3362:power set
3332:partition
3249:Predicate
3195:Predicate
3110:Syllogism
3100:Soundness
3073:Inference
3063:Tautology
2965:paradoxes
2728:Paradoxes
2648:Axiomatic
2627:Universal
2603:Singleton
2598:Recursive
2541:Countable
2536:Amorphous
2395:Power set
2312:Power set
2270:dependent
2265:countable
2170:EMS Press
2076:ProofWiki
1788:⋃
1764:∪
1696:∞
1681:⋃
1647:∈
1605:index set
1579:∈
1523:∈
1516:⋃
1485:∈
1478:⋃
1452:⋃
1405:⋃
1374:∪
1371:⋯
1368:∪
1355:∪
1342:∪
1299:…
1248:Notations
1191:∈
1174:∈
1168:∃
1164:⟺
1155:⋃
1152:∈
1118:there is
971:∁
942:∁
932:∁
924:∩
919:∁
902:∪
855:of a set
853:power set
830:∪
821:∩
812:∪
794:∩
785:∪
756:∩
747:∪
738:∩
720:∪
711:∩
670:∪
613:∅
610:∪
591:empty set
568:∪
562:∪
531:∪
522:∪
504:∪
495:∪
384:…
342:∪
266:∈
252:∈
231:∪
172:empty set
92:∪
86:∪
49:∪
18:Set union
4550:Logicism
4543:timeline
4519:Concrete
4378:Validity
4348:T-schema
4341:Kripke's
4336:Tarski's
4331:semantic
4321:Strength
4270:submodel
4265:spectrum
4233:function
4081:Tarski's
4070:Elements
4057:geometry
4013:Robinson
3934:variable
3919:function
3892:spectrum
3882:Sentence
3838:variable
3781:Language
3734:Relation
3695:Automata
3685:Alphabet
3669:language
3523:-jection
3501:codomain
3487:Function
3448:Universe
3418:Infinite
3322:Relation
3105:Validity
3095:Argument
2993:theorem,
2732:Problems
2636:Theories
2612:Superset
2588:Infinite
2417:Concepts
2297:Infinity
2221:Overview
2086:29 April
2080:Archived
1913:Archived
1808:See also
1747:∪
1599:, where
1122:element
132:elements
4492:Related
4289:Diagram
4187: (
4166:Hilbert
4151:Systems
4146:Theorem
4024:of the
3969:systems
3749:Formula
3744:Grammar
3660: (
3604:General
3317:Forcing
3302:Element
3222:Monadic
2997:paradox
2938:Theorem
2874:General
2670:General
2665:Zermelo
2571:subbase
2553: (
2492:Forcing
2470:Element
2442: (
2420:Methods
2307:Pairing
2172:, 2001
2142:Unicode
1909:"Union"
1802:\bigcup
1737:Unicode
1662:
1636:
1010:
990:
887:, is a
877:
857:
691:
659:
655:
635:
631:
599:
583:
551:
482:
445:
317:integer
166:
146:
4255:finite
4018:Skolem
3971:
3946:Theory
3914:Symbol
3904:String
3887:atomic
3764:ground
3759:closed
3754:atomic
3710:ground
3673:syntax
3569:binary
3496:domain
3413:Finite
3178:finite
3036:Logics
2995:
2943:Theory
2561:Filter
2551:Finite
2487:Family
2430:Almost
2275:global
2260:Choice
2247:Axioms
2116:
2054:
2027:
2000:
1973:
1744:
1742:U+222A
1603:is an
1507:, and
1185:
1072:, and
1028:, and
883:, and
593:is an
213:. In
144:zero (
122:, the
80:
43:
4245:Model
3993:Peano
3850:Proof
3690:Arity
3619:Naive
3506:image
3438:Fuzzy
3398:Empty
3347:union
3292:Class
2933:Model
2923:Lemma
2881:Axiom
2653:Naive
2583:Fuzzy
2546:Empty
2529:types
2480:tuple
2450:Class
2444:large
2405:Union
2322:Union
2137:(PDF)
1895:Notes
1752:. In
1750:UNION
1105:class
1099:. If
201:, in
124:union
4368:Type
4171:list
3975:list
3952:list
3941:Term
3875:rank
3769:open
3663:list
3475:Maps
3380:sets
3239:Free
3209:list
2959:list
2886:list
2566:base
2114:ISBN
2088:2018
2052:ISBN
2025:ISBN
1998:ISBN
1971:ISBN
1780:and
1778:\cup
1607:and
851:The
433:and
209:and
193:and
128:sets
4055:of
4037:of
3985:of
3517:Sur
3491:Map
3298:Ur-
3280:Set
2527:Set
1754:TeX
1735:In
1396:or
1126:of
404:not
325:= {
311:= {
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118:In
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3666:),
3521:Bi
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2168:,
2162:,
2139:.
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2078:.
2074:.
1955:^
1939:.
1928:^
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1270:,
1265:1
1261:S
1242:M
1238:M
1234:C
1230:B
1226:A
1222:C
1218:B
1214:A
1197:.
1194:A
1188:x
1182:,
1178:M
1171:A
1159:M
1149:x
1136:A
1132:x
1128:M
1124:A
1113:M
1109:x
1101:M
1074:C
1070:B
1066:A
1062:x
1058:C
1054:B
1050:A
1046:x
1042:C
1038:B
1034:A
1030:C
1026:B
1022:A
998:U
947:,
938:)
928:B
915:A
911:(
908:=
905:B
899:A
865:U
839:.
836:)
833:C
827:A
824:(
818:)
815:B
809:A
806:(
803:=
800:)
797:C
791:B
788:(
782:A
762:)
759:C
753:A
750:(
744:)
741:B
735:A
732:(
729:=
726:)
723:C
717:B
714:(
708:A
679:A
676:=
673:A
667:A
643:A
619:A
616:=
607:A
571:C
565:B
559:A
537:.
534:C
528:)
525:B
519:A
516:(
513:=
510:)
507:C
501:B
498:(
492:A
470:C
462:,
459:B
456:,
453:A
387:}
381:,
378:6
375:,
372:5
369:,
366:4
363:,
360:3
357:,
354:2
351:{
348:=
345:B
339:A
327:x
323:B
313:x
309:A
302:B
298:A
294:B
290:A
284:.
272:}
269:B
263:x
255:A
249:x
246::
243:x
240:{
237:=
234:B
228:A
211:B
207:A
203:B
199:A
195:B
191:A
168:)
154:0
95:C
89:B
83:A
52:B
46:A
20:)
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