5322:
4252:
521:
There is no order among the elements of a set (this explains and validates the equality of the last example), but with the ellipses notation, we use an ordered sequence before (or after) the ellipsis as a convenient notational vehicle for explaining which elements are in a set. The first few elements
1687:
is clear from context, it may be not explicitly specified. It is common in the literature for an author to state the domain ahead of time, and then not specify it in the set-builder notation. For example, an author may say something such as, "Unless otherwise stated, variables are to be taken to be
892:
In each preceding example, each set is described by enumerating its elements. Not all sets can be described in this way, or if they can, their enumeration may be too long or too complicated to be useful. Therefore, many sets are defined by a property that characterizes their elements. This
1331:
522:
of the sequence are shown, then the ellipses indicate that the simplest interpretation should be applied for continuing the sequence. Should no terminating value appear to the right of the ellipses, then the sequence is considered to be unbounded.
2702:
3881:
3777:
3476:
2936:
2285:
1696:
The following examples illustrate particular sets defined by set-builder notation via predicates. In each case, the domain is specified on the left side of the vertical bar, while the rule is specified on the right side.
2047:
2136:
940:
However, the prose approach may lack accuracy or be ambiguous. Thus, set-builder notation is often used with a predicate characterizing the elements of the set being defined, as described in the following section.
515:
3486:
Two sets are equal if and only if they have the same elements. Sets defined by set builder notation are equal if and only if their set builder rules, including the domain specifiers, are equivalent. That is
4449:
4291:, and set objects, respectively. Python uses an English-based syntax. Haskell replaces the set-builder's braces with square brackets and uses symbols, including the standard set-builder vertical bar.
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78:
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Therefore, in order to prove the equality of two sets defined by set builder notation, it suffices to prove the equivalence of their predicates, including the domain qualifiers.
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393:
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207:
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1407:
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969:
separator, and a predicate. Thus there is a variable on the left of the separator, and a rule on the right of it. These three parts are contained in curly brackets:
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855:
703:
934:
914:
581:
677:
3687:
3383:
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2209:
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When inverse functions can be explicitly stated, the expression on the left can be eliminated through simple substitution. Consider the example set
1951:
5779:
2052:
404:
4369:
3015:
1517:
25:
5129:
3493:
5468:
5288:
2525:
2349:
4298:
using
Sequence Comprehensions, where the "for" keyword returns a list of the yielded variables using the "yield" keyword.
1688:
natural numbers," though in less formal contexts where the domain can be assumed, a written mention is often unnecessary.
4050:
3181:
5796:
1885:
162:
can be described directly by enumerating all of its elements between curly brackets, as in the following two examples:
1703:
2317:
1794:
3246:
3130:
1412:
1678:
although seemingly well formed as a set builder expression, cannot define a set without producing a contradiction.
4287:
In Python, the set-builder's braces are replaced with square brackets, parentheses, or curly braces, giving list,
2948:
5774:
4007:
5654:
5077:
The set builder notation and list comprehension notation are both instances of a more general notation known as
2141:
5934:
5082:
4671:
4549:
4236:
4224:
950:
1179:
5548:
5427:
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1470:
2775:
1326:{\displaystyle \{x\mid x\in E{\text{ and }}\Phi (x)\}\quad {\text{or}}\quad \{x\mid x\in E\land \Phi (x)\}.}
781:
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302:
1632:
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214:
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1101:
975:
2459:
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is a notational variant for the same set of even natural numbers. It is not necessary to specify that
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5365:
5353:
5348:
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1022:
708:
153:
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for which the predicate is true. This can easily lead to contradictions and paradoxes. For example,
618:
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168:
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creates a set of pairs, where each pair puts an integer into correspondence with an odd integer.
2697:{\displaystyle \{a\in \mathbb {R} \mid (\exists p\in \mathbb {Z} )(\exists q\in \mathbb {Z} )\}}
761:
5845:
5726:
5538:
5358:
5098:
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4011:
3876:{\displaystyle (x\in \mathbb {R} \land x^{2}=1)\Leftrightarrow (x\in \mathbb {Q} \land |x|=1).}
3330:
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116:
5197:
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5483:
5370:
4010:, set builder notation is not part of the formal syntax of the theory. Instead, there is a
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1626:
833:
893:
characterization may be done informally using general prose, as in the following example.
8:
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for which the predicate holds (is true) belong to the set being defined. All values of
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656:
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When it is desired to denote a set that contains elements from a regular sequence, an
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obtained from this axiom is exactly the set described in set builder notation as
3772:{\displaystyle \{x\in \mathbb {R} \mid x^{2}=1\}=\{x\in \mathbb {Q} \mid |x|=1\}}
3471:{\displaystyle \{2t+1\mid t\in \mathbb {Z} \}=\{u\mid (u-1)/2\in \mathbb {Z} \}.}
2705:
1747:
2931:{\displaystyle \{f(x)\mid \Phi (x)\}=\{y\mid \exists x(y=f(x)\wedge \Phi (x))\}}
5903:
5700:
5681:
5585:
5570:
5527:
5463:
5405:
2431:
2280:{\displaystyle \{(x,y)\in \mathbb {R} \times \mathbb {R} \mid 0<y<f(x)\}}
1337:
396:
5923:
5908:
5710:
5624:
5619:
2042:{\displaystyle G_{m}=\{x\in \mathbb {Z} \mid x\geq m\}=\{m,m+1,m+2,\ldots \}}
961:
otherwise. In this form, set-builder notation has three parts: a variable, a
124:
5878:
4301:
Consider these set-builder notation examples in some programming languages:
2716:
5858:
5853:
5671:
5600:
5558:
5417:
5321:
5086:
966:
85:
3010:
is the set of all natural numbers, is the set of all even natural numbers.
209:
is the set containing the four numbers 3, 7, 15, and 31, and nothing else.
5883:
5518:
5172:
2428:
2131:{\displaystyle G_{3}=\{x\in \mathbb {Z} \mid x\geq 3\}=\{3,4,5,\ldots \}}
1750:
104:
949:
Set-builder notation can be used to describe a set that is defined by a
5863:
5634:
5297:
510:{\displaystyle \{\ldots ,-2,-1,0,1,2,\ldots \}=\{0,1,-1,2,-2,\ldots \}}
96:
1613:
In general, it is not a good idea to consider sets without defining a
1098:
for which the predicate does not hold do not belong to the set. Thus
5666:
5629:
5580:
5478:
1150:
756:
140:
2595:
is a natural number, as this is implied by the formula on the right.
293:
289:
This is sometimes called the "roster method" for specifying a set.
285:, and nothing else (there is no order among the elements of a set).
2721:
An extension of set-builder notation replaces the single variable
2709:
4028:
is a formula in the language of set theory, then there is a set
2708:; that is, real numbers that can be written as the ratio of two
2458:. The ∃ sign stands for "there exists", which is known as
1066:
The vertical bar (or colon) is a separator that can be read as "
5691:
5513:
4835:
1618:
5144:
Richard
Aufmann, Vernon C. Barker, and Joanne Lockwood, 2007,
5563:
5330:
5266:
5124:(6th ed.). New York, NY: McGraw-Hill. pp. 111–112.
100:
4444:{\displaystyle \{(k,x)\ |\ k\in K\wedge x\in X\wedge P(x)\}}
1936:
1070:", "for which", or "with the property that". The formula
127:, or stating the properties that its members must satisfy.
3782:
because the two rule predicates are logically equivalent:
4765:
3481:
2717:
More complex expressions on the left side of the notation
296:
notation may be employed, as shown in the next examples:
5222:
Irvine, Andrew David; Deutsch, Harry (9 October 2016) .
3071:{\displaystyle \{p/q\mid p,q\in \mathbb {Z} ,q\not =0\}}
1583:{\displaystyle \{x\in E\mid \Phi _{1}(x),\Phi _{2}(x)\}}
73:{\displaystyle \{n\mid \exists k\in \mathbb {Z} ,n=2k\}}
3563:{\displaystyle \{x\in A\mid P(x)\}=\{x\in B\mid Q(x)\}}
4372:
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2582:{\displaystyle \{n\mid (\exists k\in \mathbb {N} )\}}
2528:
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2420:{\displaystyle \{n\in \mathbb {N} \mid (\exists k)\}}
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217:
171:
28:
5081:, which permits map/filter-like operations over any
3886:
This equivalence holds because, for any real number
1364:. This notation represents the set of all values of
1360:
symbol denotes the logical "and" operator, known as
349:
is the set of integers between 1 and 100 inclusive.
4443:
4357:
4193:
4138:{\displaystyle (\forall E)(\exists Y)(\forall x).}
4137:
3990:
3955:
3915:
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3232:{\displaystyle \{(t,2t+1)\mid t\in \mathbb {Z} \}}
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607:
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387:
341:
265:
201:
72:
4259:It has been suggested that parts of this page be
2344:denotes the set of ordered pairs of real numbers.
1928:{\displaystyle \{x\in \mathbb {R} \mid x^{2}=1\}}
5921:
3963:. In particular, both sets are equal to the set
1739:{\displaystyle \{x\in \mathbb {R} \mid x>0\}}
2337:{\displaystyle \mathbb {R} \times \mathbb {R} }
1840:{\displaystyle \{x\in \mathbb {R} \mid |x|=1\}}
953:, that is, a logical formula that evaluates to
3285:{\displaystyle \{2t+1\mid t\in \mathbb {Z} \}}
3169:{\displaystyle \{2t+1\mid t\in \mathbb {Z} \}}
2287:is the set of pairs of real numbers such that
1460:{\displaystyle \Phi _{1}(x)\land \Phi _{2}(x)}
944:
147:
5282:
5221:
1173:can appear on the left of the vertical bar:
130:Defining sets by properties is also known as
4438:
4373:
4352:
4326:
4215:A similar notation available in a number of
4188:
4161:
3985:
3970:
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3730:
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3226:
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3019:
2981:{\displaystyle \{2n\mid n\in \mathbb {N} \}}
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408:
382:
358:
336:
306:
260:
242:
236:
218:
196:
172:
67:
29:
4204:
936:is the set of all addresses on Pine Street.
5289:
5275:
4034:whose members are exactly the elements of
2197:{\displaystyle G_{-2}=\{-2,-1,0,\ldots \}}
5121:Discrete Mathematics and its Applications
3841:
3802:
3740:
3701:
3458:
3412:
3275:
3222:
3159:
3108:
3086:
3049:
2996:
2971:
2654:
2634:
2614:
2551:
2454:sign stands for "and", which is known as
2392:
2363:
2330:
2322:
2243:
2235:
2079:
1978:
1899:
1808:
1717:
48:
1377:
1219:{\displaystyle \{x\in E\mid \Phi (x)\},}
1162:
154:Set (mathematics) § Roster notation
5159:A Transition to Mathematics with Proofs
4194:{\displaystyle \{x\in E\mid \Phi (x)\}}
1507:{\displaystyle \{x\in E\mid \Phi (x)\}}
563:denotes the set of all natural numbers
5922:
5161:, Jones & Bartlett, pp. 44ff.
5146:Intermediate Algebra with Applications
4001:
3482:Equivalent predicates yield equal sets
2818:{\displaystyle \{f(x)\mid \Phi (x)\},}
1937:equivalent predicates yield equal sets
1590:, using a comma instead of the symbol
1376:for which the predicate is true (see "
823:{\displaystyle \{a_{1},\dots ,a_{n}\}}
5270:
5170:
5117:
4006:In many formal set theories, such as
3377:in the set builder notation to find
1229:or by adjoining it to the predicate:
342:{\displaystyle \{1,2,3,\ldots ,100\}}
4245:
1671:{\displaystyle \{x~|~x\not \in x\},}
5228:Stanford Encyclopedia of Philosophy
266:{\displaystyle \{a,c,b\}=\{a,b,c\}}
13:
4176:
4117:
4081:
4069:
4057:
3586:
2910:
2880:
2853:
2797:
2765:{\displaystyle \{x\mid \Phi (x)\}}
2747:
2644:
2624:
2541:
2472:
2373:
1882:. This set can also be defined as
1773:
1623:all possible things that may exist
1559:
1537:
1489:
1439:
1417:
1387:
1305:
1262:
1198:
1132:{\displaystyle \{x\mid \Phi (x)\}}
1114:
1035:
1006:{\displaystyle \{x\mid \Phi (x)\}}
988:
765:
90:expressed in set-builder notation.
38:
14:
5946:
388:{\displaystyle \{1,2,3,\ldots \}}
16:Use of braces for specifying sets
5320:
4358:{\displaystyle \{l\ |\ l\in L\}}
4250:
3125:the set of all rational numbers.
2291:is greater than 0 and less than
1336:The ∈ symbol here denotes
3100:is the set of all integers, is
2496:{\displaystyle (\exists x)P(x)}
1283:
1277:
1056:{\displaystyle \{x:\Phi (x)\}.}
957:for an element of the set, and
748:{\displaystyle =\{1,\dots ,0\}}
5296:
5241:
5215:
5190:
5164:
5151:
5138:
5111:
4435:
4429:
4395:
4388:
4376:
4336:
4185:
4179:
4129:
4126:
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3943:
3935:
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3756:
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3661:
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3631:
3628:
3625:
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3616:
3598:
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3592:
3583:
3554:
3548:
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3515:
3443:
3431:
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3340:
3209:
3188:
2922:
2919:
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2898:
2886:
2862:
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2800:
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2638:
2621:
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2558:
2555:
2538:
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2469:
2411:
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2379:
2370:
2271:
2265:
2228:
2216:
1824:
1816:
1776:
1764:
1646:
1617:, as this would represent the
1574:
1568:
1552:
1546:
1498:
1492:
1454:
1448:
1432:
1426:
1396:
1390:
1370:that belong to some given set
1314:
1308:
1271:
1265:
1207:
1201:
1123:
1117:
1044:
1038:
997:
991:
718:
712:
666:
660:
646:{\displaystyle \{1,\dots ,n\}}
556:{\displaystyle \{1,\dots ,n\}}
1:
3118:{\displaystyle \mathbb {Q} ,}
2503:is read as "there exists an
882:{\displaystyle 1\leq i\leq n}
608:{\displaystyle 1\leq i\leq n}
202:{\displaystyle \{7,3,15,31\}}
4294:The same can be achieved in
4239:operations over one or more
3093:{\displaystyle \mathbb {Z} }
3003:{\displaystyle \mathbb {N} }
1139:is the set of all values of
7:
5092:
4008:Zermelo–Fraenkel set theory
3667:{\displaystyle (\forall t)}
3176:is the set of odd integers.
1782:{\displaystyle (0,\infty )}
1746:is the set of all strictly
1691:
945:Sets defined by a predicate
679:. A subtle special case is
517:is the set of all integers.
148:Sets defined by enumeration
10:
5951:
5780:von Neumann–Bernays–Gödel
5157:Michael J Cullinan, 2012,
4208:
4012:set existence axiom scheme
3927:is a rational number with
2460:existential quantification
1753:, which can be written in
1629:shows that the expression
771:{\displaystyle \emptyset }
151:
5844:
5807:
5719:
5609:
5581:One-to-one correspondence
5497:
5438:
5329:
5318:
5304:
5249:"Sequence Comprehensions"
5148:, Brooks Cole, p. 6.
4834:
4764:
4670:
4567:
4548:
4454:
3366:{\displaystyle t=(u-1)/2}
1145:that satisfy the formula
916:addresses on Pine Street
5104:
5038:
5011:
4998:
4992:
4979:
4973:
4882:
4840:
4788:
4770:
4700:
4676:
4603:
4573:
4560:
4554:
4487:
4460:
4205:In programming languages
3991:{\displaystyle \{-1,1\}}
3292:. Make the substitution
1875:{\displaystyle \{-1,1\}}
1402:{\displaystyle \Phi (x)}
653:is the bracket notation
99:and its applications to
5118:Rosen, Kenneth (2007).
4014:, which states that if
3916:{\displaystyle x^{2}=1}
1681:In cases where the set
1159:satisfies the formula.
830:denotes the set of all
615:. Another notation for
138:or as defining a set's
5539:Constructible universe
5366:Constructibility (V=L)
5198:"Set-Builder Notation"
5099:Glossary of set theory
4445:
4359:
4195:
4139:
3992:
3957:
3917:
3877:
3773:
3668:
3564:
3472:
3367:
3321:
3320:{\displaystyle u=2t+1}
3286:
3233:
3170:
3119:
3094:
3072:
3004:
2982:
2932:
2819:
2766:
2698:
2583:
2497:
2448:
2447:{\displaystyle \land }
2421:
2338:
2281:
2198:
2132:
2043:
1929:
1876:
1841:
1783:
1740:
1672:
1604:
1603:{\displaystyle \land }
1584:
1508:
1461:
1403:
1354:
1353:{\displaystyle \land }
1327:
1220:
1133:
1057:
1007:
930:
910:
883:
851:
824:
772:
749:
699:
673:
647:
609:
577:
557:
511:
389:
343:
273:is the set containing
267:
203:
81:
74:
5935:Mathematical notation
5762:Principia Mathematica
5596:Transfinite induction
5455:(i.e. set difference)
5177:mathworld.wolfram.com
4446:
4360:
4217:programming languages
4196:
4140:
3993:
3958:
3956:{\displaystyle |x|=1}
3918:
3878:
3774:
3669:
3565:
3473:
3368:
3322:
3287:
3234:
3171:
3120:
3095:
3073:
3005:
2983:
2933:
2825:which should be read
2820:
2767:
2699:
2584:
2498:
2449:
2422:
2339:
2282:
2199:
2133:
2044:
1930:
1877:
1842:
1784:
1741:
1673:
1605:
1585:
1514:is sometimes written
1509:
1462:
1404:
1355:
1328:
1221:
1163:Specifying the domain
1134:
1058:
1008:
931:
911:
884:
852:
850:{\displaystyle a_{i}}
825:
773:
750:
700:
674:
648:
610:
578:
558:
512:
390:
344:
268:
204:
117:mathematical notation
75:
21:
5836:Burali-Forti paradox
5591:Set-builder notation
5544:Continuum hypothesis
5484:Symmetric difference
5079:monad comprehensions
4370:
4323:
4158:
4051:
3967:
3931:
3894:
3789:
3688:
3580:
3494:
3384:
3331:
3296:
3247:
3182:
3131:
3104:
3082:
3016:
2992:
2949:
2832:
2776:
2735:
2601:
2526:
2466:
2438:
2350:
2318:
2210:
2142:
2053:
1952:
1886:
1851:
1795:
1761:
1704:
1633:
1594:
1518:
1471:
1413:
1384:
1344:
1236:
1180:
1102:
1023:
976:
920:
900:
861:
834:
782:
762:
709:
683:
657:
619:
587:
567:
529:
405:
355:
303:
215:
169:
113:set-builder notation
26:
5797:Tarski–Grothendieck
5224:"Russell's Paradox"
5171:Weisstein, Eric W.
4002:Set existence axiom
2456:logical conjunction
1615:domain of discourse
1378:Set existence axiom
1362:logical conjunction
698:{\displaystyle n=0}
123:by enumerating its
5386:Limitation of size
4441:
4355:
4265:List comprehension
4229:list comprehension
4211:List comprehension
4191:
4135:
3988:
3953:
3913:
3873:
3769:
3664:
3560:
3468:
3363:
3327:, which is to say
3317:
3282:
3229:
3166:
3115:
3090:
3068:
3000:
2978:
2928:
2815:
2762:
2694:
2579:
2493:
2462:. So for example,
2444:
2427:is the set of all
2417:
2334:
2277:
2194:
2128:
2039:
1925:
1872:
1837:
1779:
1736:
1668:
1600:
1580:
1504:
1457:
1399:
1350:
1323:
1216:
1129:
1078:is said to be the
1053:
1003:
929:{\displaystyle \}}
926:
909:{\displaystyle \{}
906:
879:
847:
820:
768:
745:
695:
669:
643:
605:
573:
553:
507:
385:
339:
263:
199:
70:
5917:
5916:
5826:Russell's paradox
5775:Zermelo–Fraenkel
5676:Dedekind-infinite
5549:Diagonal argument
5448:Cartesian product
5312:Set (mathematics)
5131:978-0-07-288008-3
5075:
5074:
4401:
4393:
4342:
4334:
4285:
4284:
4231:, which combines
2731:. So instead of
2313:cartesian product
2049:. As an example,
1942:For each integer
1755:interval notation
1652:
1644:
1627:Russell's paradox
1409:is a conjunction
1281:
1260:
1153:, if no value of
576:{\displaystyle i}
132:set comprehension
119:for describing a
5942:
5899:Bertrand Russell
5889:John von Neumann
5874:Abraham Fraenkel
5869:Richard Dedekind
5831:Suslin's problem
5742:Cantor's theorem
5459:De Morgan's laws
5324:
5291:
5284:
5277:
5268:
5267:
5261:
5260:
5258:
5256:
5245:
5239:
5238:
5236:
5234:
5219:
5213:
5212:
5210:
5208:
5194:
5188:
5187:
5185:
5183:
5168:
5162:
5155:
5149:
5142:
5136:
5135:
5115:
5069:
5066:
5063:
5060:
5057:
5054:
5051:
5048:
5045:
5042:
5033:
5030:
5027:
5024:
5021:
5018:
5015:
4961:
4958:
4955:
4952:
4949:
4946:
4943:
4940:
4937:
4934:
4931:
4928:
4925:
4922:
4919:
4916:
4913:
4910:
4907:
4904:
4901:
4898:
4895:
4892:
4889:
4886:
4877:
4874:
4871:
4868:
4865:
4862:
4859:
4856:
4853:
4850:
4847:
4844:
4828:
4825:
4822:
4819:
4816:
4813:
4810:
4807:
4804:
4801:
4798:
4795:
4792:
4783:
4780:
4777:
4774:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4695:
4692:
4689:
4686:
4683:
4680:
4664:
4661:
4658:
4655:
4652:
4649:
4646:
4643:
4640:
4637:
4634:
4631:
4628:
4625:
4622:
4619:
4616:
4613:
4610:
4607:
4598:
4595:
4592:
4589:
4586:
4583:
4580:
4577:
4542:
4539:
4536:
4533:
4530:
4527:
4524:
4521:
4518:
4515:
4512:
4509:
4506:
4503:
4500:
4497:
4494:
4491:
4482:
4479:
4476:
4473:
4470:
4467:
4464:
4450:
4448:
4447:
4442:
4399:
4398:
4391:
4364:
4362:
4361:
4356:
4340:
4339:
4332:
4304:
4303:
4280:
4277:
4254:
4253:
4246:
4200:
4198:
4197:
4192:
4153:
4144:
4142:
4141:
4136:
4043:
4039:
4033:
4027:
4019:
3997:
3995:
3994:
3989:
3962:
3960:
3959:
3954:
3946:
3938:
3922:
3920:
3919:
3914:
3906:
3905:
3882:
3880:
3879:
3874:
3860:
3852:
3844:
3818:
3817:
3805:
3778:
3776:
3775:
3770:
3759:
3751:
3743:
3717:
3716:
3704:
3673:
3671:
3670:
3665:
3573:if and only if
3569:
3567:
3566:
3561:
3477:
3475:
3474:
3469:
3461:
3450:
3415:
3372:
3370:
3369:
3364:
3359:
3326:
3324:
3323:
3318:
3291:
3289:
3288:
3283:
3278:
3238:
3236:
3235:
3230:
3225:
3175:
3173:
3172:
3167:
3162:
3124:
3122:
3121:
3116:
3111:
3099:
3097:
3096:
3091:
3089:
3077:
3075:
3074:
3069:
3052:
3029:
3009:
3007:
3006:
3001:
2999:
2987:
2985:
2984:
2979:
2974:
2937:
2935:
2934:
2929:
2824:
2822:
2821:
2816:
2771:
2769:
2768:
2763:
2726:
2706:rational numbers
2703:
2701:
2700:
2695:
2657:
2637:
2617:
2594:
2588:
2586:
2585:
2580:
2554:
2519:
2508:
2502:
2500:
2499:
2494:
2453:
2451:
2450:
2445:
2426:
2424:
2423:
2418:
2395:
2366:
2343:
2341:
2340:
2335:
2333:
2325:
2310:
2301:
2286:
2284:
2283:
2278:
2246:
2238:
2203:
2201:
2200:
2195:
2157:
2156:
2137:
2135:
2134:
2129:
2082:
2065:
2064:
2048:
2046:
2045:
2040:
1981:
1964:
1963:
1948:, we can define
1947:
1934:
1932:
1931:
1926:
1915:
1914:
1902:
1881:
1879:
1878:
1873:
1846:
1844:
1843:
1838:
1827:
1819:
1811:
1788:
1786:
1785:
1780:
1745:
1743:
1742:
1737:
1720:
1686:
1677:
1675:
1674:
1669:
1650:
1649:
1642:
1609:
1607:
1606:
1601:
1589:
1587:
1586:
1581:
1567:
1566:
1545:
1544:
1513:
1511:
1510:
1505:
1466:
1464:
1463:
1458:
1447:
1446:
1425:
1424:
1408:
1406:
1405:
1400:
1375:
1369:
1359:
1357:
1356:
1351:
1332:
1330:
1329:
1324:
1282:
1279:
1261:
1258:
1225:
1223:
1222:
1217:
1172:
1158:
1149:. It may be the
1148:
1144:
1138:
1136:
1135:
1130:
1097:
1091:
1086:. All values of
1077:
1062:
1060:
1059:
1054:
1012:
1010:
1009:
1004:
935:
933:
932:
927:
915:
913:
912:
907:
888:
886:
885:
880:
856:
854:
853:
848:
846:
845:
829:
827:
826:
821:
816:
815:
797:
796:
777:
775:
774:
769:
755:is equal to the
754:
752:
751:
746:
704:
702:
701:
696:
678:
676:
675:
672:{\displaystyle }
670:
652:
650:
649:
644:
614:
612:
611:
606:
582:
580:
579:
574:
562:
560:
559:
554:
516:
514:
513:
508:
394:
392:
391:
386:
348:
346:
345:
340:
284:
280:
276:
272:
270:
269:
264:
208:
206:
205:
200:
109:computer science
91:
79:
77:
76:
71:
51:
5950:
5949:
5945:
5944:
5943:
5941:
5940:
5939:
5920:
5919:
5918:
5913:
5840:
5819:
5803:
5768:New Foundations
5715:
5605:
5524:Cardinal number
5507:
5493:
5434:
5325:
5316:
5300:
5295:
5265:
5264:
5254:
5252:
5247:
5246:
5242:
5232:
5230:
5220:
5216:
5206:
5204:
5196:
5195:
5191:
5181:
5179:
5169:
5165:
5156:
5152:
5143:
5139:
5132:
5116:
5112:
5107:
5095:
5071:
5070:
5067:
5064:
5061:
5058:
5055:
5052:
5049:
5046:
5043:
5040:
5035:
5034:
5031:
5028:
5025:
5022:
5019:
5016:
5013:
5001:
5000:
4995:
4994:
4982:
4981:
4976:
4975:
4963:
4962:
4959:
4956:
4953:
4950:
4947:
4944:
4941:
4938:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4879:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4830:
4829:
4826:
4823:
4820:
4817:
4814:
4811:
4808:
4805:
4802:
4799:
4796:
4793:
4790:
4785:
4784:
4781:
4778:
4775:
4772:
4760:
4759:
4756:
4753:
4750:
4747:
4744:
4741:
4738:
4735:
4732:
4729:
4726:
4723:
4720:
4717:
4714:
4711:
4708:
4705:
4702:
4697:
4696:
4693:
4690:
4687:
4684:
4681:
4678:
4666:
4665:
4662:
4659:
4656:
4653:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4629:
4626:
4623:
4620:
4617:
4614:
4611:
4608:
4605:
4600:
4599:
4596:
4593:
4590:
4587:
4584:
4581:
4578:
4575:
4563:
4562:
4557:
4556:
4544:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4522:
4519:
4516:
4513:
4510:
4507:
4504:
4501:
4498:
4495:
4492:
4489:
4484:
4483:
4480:
4477:
4474:
4471:
4468:
4465:
4462:
4394:
4371:
4368:
4367:
4335:
4324:
4321:
4320:
4281:
4275:
4272:
4255:
4251:
4213:
4207:
4159:
4156:
4155:
4149:
4052:
4049:
4048:
4041:
4035:
4029:
4021:
4015:
4004:
3968:
3965:
3964:
3942:
3934:
3932:
3929:
3928:
3923:if and only if
3901:
3897:
3895:
3892:
3891:
3856:
3848:
3840:
3813:
3809:
3801:
3790:
3787:
3786:
3755:
3747:
3739:
3712:
3708:
3700:
3689:
3686:
3685:
3581:
3578:
3577:
3495:
3492:
3491:
3484:
3457:
3446:
3411:
3385:
3382:
3381:
3373:, then replace
3355:
3332:
3329:
3328:
3297:
3294:
3293:
3274:
3248:
3245:
3244:
3221:
3183:
3180:
3179:
3158:
3132:
3129:
3128:
3107:
3105:
3102:
3101:
3085:
3083:
3080:
3079:
3048:
3025:
3017:
3014:
3013:
2995:
2993:
2990:
2989:
2970:
2950:
2947:
2946:
2833:
2830:
2829:
2777:
2774:
2773:
2736:
2733:
2732:
2722:
2719:
2653:
2633:
2613:
2602:
2599:
2598:
2590:
2550:
2527:
2524:
2523:
2510:
2504:
2467:
2464:
2463:
2439:
2436:
2435:
2432:natural numbers
2391:
2362:
2351:
2348:
2347:
2329:
2321:
2319:
2316:
2315:
2306:
2292:
2242:
2234:
2211:
2208:
2207:
2149:
2145:
2143:
2140:
2139:
2078:
2060:
2056:
2054:
2051:
2050:
1977:
1959:
1955:
1953:
1950:
1949:
1943:
1910:
1906:
1898:
1887:
1884:
1883:
1852:
1849:
1848:
1823:
1815:
1807:
1796:
1793:
1792:
1762:
1759:
1758:
1716:
1705:
1702:
1701:
1694:
1682:
1645:
1634:
1631:
1630:
1595:
1592:
1591:
1562:
1558:
1540:
1536:
1519:
1516:
1515:
1472:
1469:
1468:
1442:
1438:
1420:
1416:
1414:
1411:
1410:
1385:
1382:
1381:
1371:
1365:
1345:
1342:
1341:
1278:
1259: and
1257:
1237:
1234:
1233:
1181:
1178:
1177:
1168:
1165:
1154:
1146:
1140:
1103:
1100:
1099:
1093:
1087:
1071:
1024:
1021:
1020:
977:
974:
973:
947:
921:
918:
917:
901:
898:
897:
862:
859:
858:
841:
837:
835:
832:
831:
811:
807:
792:
788:
783:
780:
779:
763:
760:
759:
710:
707:
706:
684:
681:
680:
658:
655:
654:
620:
617:
616:
588:
585:
584:
568:
565:
564:
530:
527:
526:
406:
403:
402:
397:natural numbers
356:
353:
352:
304:
301:
300:
282:
278:
274:
216:
213:
212:
170:
167:
166:
156:
150:
136:set abstraction
93:
89:
84:The set of all
83:
47:
27:
24:
23:
17:
12:
11:
5:
5948:
5938:
5937:
5932:
5915:
5914:
5912:
5911:
5906:
5904:Thoralf Skolem
5901:
5896:
5891:
5886:
5881:
5876:
5871:
5866:
5861:
5856:
5850:
5848:
5842:
5841:
5839:
5838:
5833:
5828:
5822:
5820:
5818:
5817:
5814:
5808:
5805:
5804:
5802:
5801:
5800:
5799:
5794:
5789:
5788:
5787:
5772:
5771:
5770:
5758:
5757:
5756:
5745:
5744:
5739:
5734:
5729:
5723:
5721:
5717:
5716:
5714:
5713:
5708:
5703:
5698:
5689:
5684:
5679:
5669:
5664:
5663:
5662:
5657:
5652:
5642:
5632:
5627:
5622:
5616:
5614:
5607:
5606:
5604:
5603:
5598:
5593:
5588:
5586:Ordinal number
5583:
5578:
5573:
5568:
5567:
5566:
5561:
5551:
5546:
5541:
5536:
5531:
5521:
5516:
5510:
5508:
5506:
5505:
5502:
5498:
5495:
5494:
5492:
5491:
5486:
5481:
5476:
5471:
5466:
5464:Disjoint union
5461:
5456:
5450:
5444:
5442:
5436:
5435:
5433:
5432:
5431:
5430:
5425:
5414:
5413:
5411:Martin's axiom
5408:
5403:
5398:
5393:
5388:
5383:
5378:
5376:Extensionality
5373:
5368:
5363:
5362:
5361:
5356:
5351:
5341:
5335:
5333:
5327:
5326:
5319:
5317:
5315:
5314:
5308:
5306:
5302:
5301:
5294:
5293:
5286:
5279:
5271:
5263:
5262:
5240:
5214:
5202:mathsisfun.com
5189:
5163:
5150:
5137:
5130:
5109:
5108:
5106:
5103:
5102:
5101:
5094:
5091:
5073:
5072:
5039:
5036:
5012:
5009:
5003:
5002:
4999:
4996:
4993:
4990:
4984:
4983:
4980:
4977:
4974:
4971:
4965:
4964:
4883:
4880:
4841:
4838:
4832:
4831:
4789:
4786:
4771:
4768:
4762:
4761:
4701:
4698:
4677:
4674:
4668:
4667:
4604:
4601:
4574:
4571:
4565:
4564:
4561:
4558:
4555:
4552:
4546:
4545:
4488:
4485:
4461:
4458:
4452:
4451:
4440:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4404:
4397:
4390:
4387:
4384:
4381:
4378:
4375:
4365:
4354:
4351:
4348:
4345:
4338:
4331:
4328:
4318:
4314:
4313:
4310:
4307:
4283:
4282:
4258:
4256:
4249:
4209:Main article:
4206:
4203:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4163:
4146:
4145:
4134:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4101:
4098:
4095:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4003:
4000:
3987:
3984:
3981:
3978:
3975:
3972:
3952:
3949:
3945:
3941:
3937:
3912:
3909:
3904:
3900:
3884:
3883:
3872:
3869:
3866:
3863:
3859:
3855:
3851:
3847:
3843:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3816:
3812:
3808:
3804:
3800:
3797:
3794:
3780:
3779:
3768:
3765:
3762:
3758:
3754:
3750:
3746:
3742:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3715:
3711:
3707:
3703:
3699:
3696:
3693:
3676:
3675:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3571:
3570:
3559:
3556:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3483:
3480:
3479:
3478:
3467:
3464:
3460:
3456:
3453:
3449:
3445:
3442:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3414:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3362:
3358:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3316:
3313:
3310:
3307:
3304:
3301:
3281:
3277:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3241:
3240:
3228:
3224:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3177:
3165:
3161:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3126:
3114:
3110:
3088:
3067:
3064:
3061:
3058:
3055:
3051:
3047:
3044:
3041:
3038:
3035:
3032:
3028:
3024:
3021:
3011:
2998:
2977:
2973:
2969:
2966:
2963:
2960:
2957:
2954:
2940:
2939:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2837:
2814:
2811:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2772:, we may have
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2718:
2715:
2714:
2713:
2704:is the set of
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2656:
2652:
2649:
2646:
2643:
2640:
2636:
2632:
2629:
2626:
2623:
2620:
2616:
2612:
2609:
2606:
2596:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2553:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2521:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2443:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2394:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2365:
2361:
2358:
2355:
2345:
2332:
2328:
2324:
2302:, for a given
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2245:
2241:
2237:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2205:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2155:
2152:
2148:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2081:
2077:
2074:
2071:
2068:
2063:
2059:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1980:
1976:
1973:
1970:
1967:
1962:
1958:
1940:
1924:
1921:
1918:
1913:
1909:
1905:
1901:
1897:
1894:
1891:
1871:
1868:
1865:
1862:
1859:
1856:
1836:
1833:
1830:
1826:
1822:
1818:
1814:
1810:
1806:
1803:
1800:
1790:
1778:
1775:
1772:
1769:
1766:
1735:
1732:
1729:
1726:
1723:
1719:
1715:
1712:
1709:
1693:
1690:
1667:
1664:
1661:
1658:
1655:
1648:
1641:
1638:
1599:
1579:
1576:
1573:
1570:
1565:
1561:
1557:
1554:
1551:
1548:
1543:
1539:
1535:
1532:
1529:
1526:
1523:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1456:
1453:
1450:
1445:
1441:
1437:
1434:
1431:
1428:
1423:
1419:
1398:
1395:
1392:
1389:
1349:
1338:set membership
1334:
1333:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1276:
1273:
1270:
1267:
1264:
1256:
1253:
1250:
1247:
1244:
1241:
1227:
1226:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1164:
1161:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1064:
1063:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1014:
1013:
1002:
999:
996:
993:
990:
987:
984:
981:
946:
943:
938:
937:
925:
905:
878:
875:
872:
869:
866:
844:
840:
819:
814:
810:
806:
803:
800:
795:
791:
787:
767:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
694:
691:
688:
668:
665:
662:
642:
639:
636:
633:
630:
627:
624:
604:
601:
598:
595:
592:
572:
552:
549:
546:
543:
540:
537:
534:
519:
518:
506:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
473:
470:
467:
464:
461:
458:
455:
452:
449:
446:
443:
440:
437:
434:
431:
428:
425:
422:
419:
416:
413:
410:
400:
395:is the set of
384:
381:
378:
375:
372:
369:
366:
363:
360:
350:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
287:
286:
262:
259:
256:
253:
250:
247:
244:
241:
238:
235:
232:
229:
226:
223:
220:
210:
198:
195:
192:
189:
186:
183:
180:
177:
174:
152:Main article:
149:
146:
69:
66:
63:
60:
57:
54:
50:
46:
43:
40:
37:
34:
31:
20:
15:
9:
6:
4:
3:
2:
5947:
5936:
5933:
5931:
5928:
5927:
5925:
5910:
5909:Ernst Zermelo
5907:
5905:
5902:
5900:
5897:
5895:
5894:Willard Quine
5892:
5890:
5887:
5885:
5882:
5880:
5877:
5875:
5872:
5870:
5867:
5865:
5862:
5860:
5857:
5855:
5852:
5851:
5849:
5847:
5846:Set theorists
5843:
5837:
5834:
5832:
5829:
5827:
5824:
5823:
5821:
5815:
5813:
5810:
5809:
5806:
5798:
5795:
5793:
5792:Kripke–Platek
5790:
5786:
5783:
5782:
5781:
5778:
5777:
5776:
5773:
5769:
5766:
5765:
5764:
5763:
5759:
5755:
5752:
5751:
5750:
5747:
5746:
5743:
5740:
5738:
5735:
5733:
5730:
5728:
5725:
5724:
5722:
5718:
5712:
5709:
5707:
5704:
5702:
5699:
5697:
5695:
5690:
5688:
5685:
5683:
5680:
5677:
5673:
5670:
5668:
5665:
5661:
5658:
5656:
5653:
5651:
5648:
5647:
5646:
5643:
5640:
5636:
5633:
5631:
5628:
5626:
5623:
5621:
5618:
5617:
5615:
5612:
5608:
5602:
5599:
5597:
5594:
5592:
5589:
5587:
5584:
5582:
5579:
5577:
5574:
5572:
5569:
5565:
5562:
5560:
5557:
5556:
5555:
5552:
5550:
5547:
5545:
5542:
5540:
5537:
5535:
5532:
5529:
5525:
5522:
5520:
5517:
5515:
5512:
5511:
5509:
5503:
5500:
5499:
5496:
5490:
5487:
5485:
5482:
5480:
5477:
5475:
5472:
5470:
5467:
5465:
5462:
5460:
5457:
5454:
5451:
5449:
5446:
5445:
5443:
5441:
5437:
5429:
5428:specification
5426:
5424:
5421:
5420:
5419:
5416:
5415:
5412:
5409:
5407:
5404:
5402:
5399:
5397:
5394:
5392:
5389:
5387:
5384:
5382:
5379:
5377:
5374:
5372:
5369:
5367:
5364:
5360:
5357:
5355:
5352:
5350:
5347:
5346:
5345:
5342:
5340:
5337:
5336:
5334:
5332:
5328:
5323:
5313:
5310:
5309:
5307:
5303:
5299:
5292:
5287:
5285:
5280:
5278:
5273:
5272:
5269:
5250:
5244:
5229:
5225:
5218:
5203:
5199:
5193:
5178:
5174:
5167:
5160:
5154:
5147:
5141:
5133:
5127:
5123:
5122:
5114:
5110:
5100:
5097:
5096:
5090:
5088:
5084:
5080:
5037:
5010:
5008:
5005:
5004:
4997:
4991:
4989:
4986:
4985:
4978:
4972:
4970:
4967:
4966:
4881:
4839:
4837:
4833:
4787:
4769:
4767:
4763:
4699:
4675:
4673:
4669:
4602:
4572:
4570:
4566:
4559:
4553:
4551:
4547:
4486:
4459:
4457:
4453:
4432:
4426:
4423:
4420:
4417:
4414:
4411:
4408:
4405:
4402:
4385:
4382:
4379:
4366:
4349:
4346:
4343:
4329:
4319:
4316:
4315:
4311:
4308:
4306:
4305:
4302:
4299:
4297:
4292:
4290:
4279:
4276:December 2023
4270:
4266:
4262:
4257:
4248:
4247:
4244:
4242:
4238:
4234:
4230:
4226:
4222:
4218:
4212:
4202:
4182:
4173:
4170:
4167:
4164:
4152:
4132:
4123:
4114:
4111:
4108:
4105:
4099:
4096:
4093:
4084:
4072:
4060:
4047:
4046:
4045:
4040:that satisfy
4038:
4032:
4025:
4020:is a set and
4018:
4013:
4009:
3999:
3982:
3979:
3976:
3973:
3950:
3947:
3939:
3926:
3910:
3907:
3902:
3898:
3889:
3870:
3864:
3861:
3853:
3845:
3837:
3834:
3822:
3819:
3814:
3810:
3806:
3798:
3795:
3785:
3784:
3783:
3763:
3760:
3752:
3744:
3736:
3733:
3727:
3721:
3718:
3713:
3709:
3705:
3697:
3694:
3684:
3683:
3682:
3681:For example,
3679:
3652:
3646:
3643:
3640:
3637:
3634:
3619:
3613:
3610:
3607:
3604:
3601:
3589:
3576:
3575:
3574:
3551:
3545:
3542:
3539:
3536:
3533:
3527:
3518:
3512:
3509:
3506:
3503:
3500:
3490:
3489:
3488:
3465:
3454:
3451:
3447:
3440:
3437:
3434:
3428:
3425:
3419:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3380:
3379:
3378:
3376:
3360:
3356:
3349:
3346:
3343:
3337:
3334:
3314:
3311:
3308:
3305:
3302:
3299:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3218:
3215:
3212:
3206:
3203:
3200:
3197:
3194:
3191:
3178:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3127:
3112:
3062:
3059:
3056:
3053:
3045:
3042:
3039:
3036:
3033:
3030:
3026:
3022:
3012:
2967:
2964:
2961:
2958:
2955:
2945:
2944:
2943:
2942:For example:
2916:
2907:
2901:
2895:
2892:
2889:
2883:
2877:
2874:
2868:
2859:
2850:
2844:
2838:
2828:
2827:
2826:
2812:
2803:
2794:
2788:
2782:
2753:
2744:
2741:
2730:
2725:
2711:
2707:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2650:
2647:
2630:
2627:
2618:
2610:
2607:
2597:
2593:
2570:
2567:
2564:
2561:
2547:
2544:
2535:
2532:
2522:
2517:
2513:
2507:
2487:
2481:
2475:
2461:
2457:
2441:
2433:
2430:
2408:
2405:
2402:
2399:
2396:
2388:
2385:
2376:
2367:
2359:
2356:
2346:
2326:
2314:
2311:. Here the
2309:
2305:
2299:
2295:
2290:
2268:
2262:
2259:
2256:
2253:
2250:
2247:
2239:
2231:
2225:
2222:
2219:
2206:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2158:
2153:
2150:
2146:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2098:
2092:
2089:
2086:
2083:
2075:
2072:
2066:
2061:
2057:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
1997:
1991:
1988:
1985:
1982:
1974:
1971:
1965:
1960:
1956:
1946:
1941:
1938:
1919:
1916:
1911:
1907:
1903:
1895:
1892:
1866:
1863:
1860:
1857:
1831:
1828:
1820:
1812:
1804:
1801:
1791:
1770:
1767:
1756:
1752:
1749:
1730:
1727:
1724:
1721:
1713:
1710:
1700:
1699:
1698:
1689:
1685:
1679:
1665:
1659:
1656:
1653:
1639:
1628:
1624:
1620:
1616:
1611:
1597:
1571:
1563:
1555:
1549:
1541:
1533:
1530:
1527:
1524:
1495:
1486:
1483:
1480:
1477:
1451:
1443:
1435:
1429:
1421:
1393:
1380:" below). If
1379:
1374:
1368:
1363:
1347:
1339:
1320:
1311:
1302:
1299:
1296:
1293:
1290:
1287:
1268:
1254:
1251:
1248:
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1242:
1232:
1231:
1230:
1213:
1204:
1195:
1192:
1189:
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1175:
1174:
1171:
1160:
1157:
1152:
1143:
1120:
1111:
1108:
1096:
1090:
1085:
1081:
1075:
1069:
1050:
1041:
1032:
1029:
1019:
1018:
1017:
994:
985:
982:
972:
971:
970:
968:
964:
960:
956:
952:
942:
896:
895:
894:
890:
876:
873:
870:
867:
864:
842:
838:
812:
808:
804:
801:
798:
793:
789:
778:. Similarly,
758:
739:
736:
733:
730:
727:
721:
715:
692:
689:
686:
663:
637:
634:
631:
628:
625:
602:
599:
596:
593:
590:
570:
547:
544:
541:
538:
535:
523:
501:
498:
495:
492:
489:
486:
483:
480:
477:
474:
471:
468:
465:
459:
453:
450:
447:
444:
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438:
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432:
429:
426:
423:
420:
417:
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411:
401:
398:
379:
376:
373:
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364:
361:
351:
333:
330:
327:
324:
321:
318:
315:
312:
309:
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298:
297:
295:
290:
257:
254:
251:
248:
245:
239:
233:
230:
227:
224:
221:
211:
193:
190:
187:
184:
181:
178:
175:
165:
164:
163:
161:
155:
145:
143:
142:
137:
133:
128:
126:
122:
118:
114:
110:
106:
102:
98:
92:
87:
86:even integers
80:
64:
61:
58:
55:
52:
44:
41:
35:
32:
19:
5859:Georg Cantor
5854:Paul Bernays
5785:Morse–Kelley
5760:
5693:
5692:Subset
5639:hereditarily
5601:Venn diagram
5590:
5559:ordered pair
5474:Intersection
5418:Axiom schema
5253:. Retrieved
5243:
5231:. Retrieved
5227:
5217:
5205:. Retrieved
5201:
5192:
5180:. Retrieved
5176:
5166:
5158:
5153:
5145:
5140:
5120:
5113:
5087:zero element
5078:
5076:
4317:Set-builder
4300:
4293:
4286:
4273:
4214:
4150:
4147:
4036:
4030:
4023:
4016:
4005:
3924:
3887:
3885:
3781:
3680:
3677:
3572:
3485:
3374:
3242:
2941:
2723:
2720:
2591:
2515:
2511:
2505:
2307:
2297:
2293:
2288:
1944:
1751:real numbers
1695:
1683:
1680:
1622:
1612:
1372:
1366:
1340:, while the
1335:
1228:
1169:
1166:
1155:
1141:
1094:
1088:
1083:
1079:
1073:
1067:
1065:
1015:
967:vertical bar
958:
954:
948:
939:
891:
525:In general,
524:
520:
291:
288:
157:
139:
135:
131:
129:
112:
94:
82:
22:
18:
5884:Thomas Jech
5727:Alternative
5706:Uncountable
5660:Ultrafilter
5519:Cardinality
5423:replacement
5371:Determinacy
5007:Mathematica
1847:is the set
705:, in which
105:mathematics
5930:Set theory
5924:Categories
5879:Kurt Gödel
5864:Paul Cohen
5701:Transitive
5469:Identities
5453:Complement
5440:Operations
5401:Regularity
5339:Adjunction
5298:Set theory
4312:Example 2
3890:, we have
2729:expression
2509:such that
583:such that
97:set theory
5812:Paradoxes
5732:Axiomatic
5711:Universal
5687:Singleton
5682:Recursive
5625:Countable
5620:Amorphous
5479:Power set
5396:Power set
5354:dependent
5349:countable
5207:20 August
5182:20 August
4424:∧
4418:∈
4412:∧
4406:∈
4347:∈
4309:Example 1
4289:generator
4227:) is the
4219:(notably
4177:Φ
4174:∣
4168:∈
4118:Φ
4115:∧
4109:∈
4103:⇔
4097:∈
4082:∀
4070:∃
4058:∀
3974:−
3846:∧
3838:∈
3829:⇔
3807:∧
3799:∈
3745:∣
3737:∈
3706:∣
3698:∈
3644:∧
3638:∈
3629:⇔
3611:∧
3605:∈
3587:∀
3543:∣
3537:∈
3510:∣
3504:∈
3455:∈
3438:−
3429:∣
3409:∈
3403:∣
3347:−
3272:∈
3266:∣
3219:∈
3213:∣
3156:∈
3150:∣
3046:∈
3034:∣
2968:∈
2962:∣
2911:Φ
2908:∧
2881:∃
2878:∣
2854:Φ
2851:∣
2798:Φ
2795:∣
2748:Φ
2745:∣
2674:∧
2651:∈
2645:∃
2631:∈
2625:∃
2619:∣
2611:∈
2548:∈
2542:∃
2536:∣
2473:∃
2442:∧
2397:∧
2389:∈
2374:∃
2368:∣
2360:∈
2327:×
2248:∣
2240:×
2232:∈
2189:…
2174:−
2165:−
2151:−
2123:…
2090:≥
2084:∣
2076:∈
2034:…
1989:≥
1983:∣
1975:∈
1904:∣
1896:∈
1858:−
1813:∣
1805:∈
1774:∞
1722:∣
1714:∈
1598:∧
1560:Φ
1538:Φ
1534:∣
1528:∈
1490:Φ
1487:∣
1481:∈
1440:Φ
1436:∧
1418:Φ
1388:Φ
1348:∧
1306:Φ
1303:∧
1297:∈
1291:∣
1263:Φ
1252:∈
1246:∣
1199:Φ
1196:∣
1190:∈
1167:A domain
1151:empty set
1115:Φ
1112:∣
1084:predicate
1068:such that
1036:Φ
989:Φ
986:∣
951:predicate
874:≤
868:≤
802:…
766:∅
757:empty set
734:…
632:…
600:≤
594:≤
542:…
502:…
493:−
478:−
454:…
427:−
418:−
412:…
380:…
328:…
141:intension
45:∈
39:∃
36:∣
5816:Problems
5720:Theories
5696:Superset
5672:Infinite
5501:Concepts
5381:Infinity
5305:Overview
5255:6 August
5233:6 August
5093:See also
4148:The set
3078:, where
3060:≠
2988:, where
2727:with an
2710:integers
2668:≠
2304:function
1748:positive
1692:Examples
1657:∉
294:ellipsis
125:elements
5754:General
5749:Zermelo
5655:subbase
5637: (
5576:Forcing
5554:Element
5526: (
5504:Methods
5391:Pairing
5251:. Scala
5085:with a
4550:Haskell
4269:Discuss
4225:Haskell
1467:, then
1082:or the
1072:Φ(
5645:Filter
5635:Finite
5571:Family
5514:Almost
5359:global
5344:Choice
5331:Axioms
5128:
5020:|->
4969:Erlang
4957:Result
4921:member
4903:member
4873:Result
4855:member
4836:Prolog
4791:SELECT
4773:SELECT
4742:select
4691:select
4456:Python
4400:
4392:
4341:
4333:
4237:filter
4221:Python
2434:. The
1939:below.
1935:; see
1651:
1643:
1619:subset
1147:Φ
281:, and
107:, and
5737:Naive
5667:Fuzzy
5630:Empty
5613:types
5564:tuple
5534:Class
5528:large
5489:Union
5406:Union
5173:"Set"
5105:Notes
5083:monad
5041:Cases
4988:Julia
4885:setof
4843:setof
4815:WHERE
4812:X_set
4806:K_set
4782:L_set
4727:where
4648:yield
4627:<-
4615:<-
4594:yield
4585:<-
4569:Scala
4296:Scala
4263:into
4261:moved
4241:lists
963:colon
959:false
115:is a
101:logic
5650:base
5257:2017
5235:2017
5209:2020
5184:2020
5126:ISBN
4939:call
4803:FROM
4779:FROM
4715:from
4703:from
4679:from
4235:and
4223:and
2429:even
2260:<
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2138:and
1728:>
1080:rule
955:true
857:for
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4900:),(
4766:SQL
4606:for
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4267:. (
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1757:as
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1016:or
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2019:,
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591:1
571:i
551:}
548:n
545:,
539:,
536:1
533:{
505:}
499:,
496:2
490:,
487:2
484:,
481:1
475:,
472:1
469:,
466:0
463:{
460:=
457:}
451:,
448:2
445:,
442:1
439:,
436:0
433:,
430:1
424:,
421:2
415:,
409:{
399:.
383:}
377:,
374:3
371:,
368:2
365:,
362:1
359:{
337:}
331:,
325:,
322:3
319:,
316:2
313:,
310:1
307:{
283:c
279:b
275:a
261:}
258:c
255:,
252:b
249:,
246:a
243:{
240:=
237:}
234:b
231:,
228:c
225:,
222:a
219:{
197:}
191:,
185:,
182:3
179:,
176:7
173:{
68:}
65:k
62:2
59:=
56:n
53:,
49:Z
42:k
33:n
30:{
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