5333:
4263:
532:
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
1698:
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
903:
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
1342:
533:
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.
2713:
3892:
3788:
3487:
2947:
2296:
1707:
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.
2058:
2147:
951:
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.
526:
3497:
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
4460:
4302:, 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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1599:
89:
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2213:
1235:
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839:
358:
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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.
1246:
282:
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404:
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3134:
898:
624:
218:
3109:
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787:
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4007:
1891:
1418:
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1619:
1369:
980:
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:
3972:
866:
714:
945:
925:
592:
688:
3698:
3394:
2842:
2220:
3254:
When inverse functions can be explicitly stated, the expression on the left can be eliminated through simple substitution. Consider the example set
1962:
5797:
2063:
415:
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3026:
1528:
36:
5140:
3504:
5486:
5299:
2536:
2360:
17:
4309:
using
Sequence Comprehensions, where the "for" keyword returns a list of the yielded variables using the "yield" keyword.
1699:
natural numbers," though in less formal contexts where the domain can be assumed, a written mention is often unnecessary.
4061:
3192:
5814:
1896:
173:
can be described directly by enumerating all of its elements between curly brackets, as in the following two examples:
1714:
2328:
1805:
3257:
3141:
1423:
1689:
although seemingly well formed as a set builder expression, cannot define a set without producing a contradiction.
4298:
In Python, the set-builder's braces are replaced with square brackets, parentheses, or curly braces, giving list,
2959:
5792:
5386:
4018:
5672:
5088:
The set builder notation and list comprehension notation are both instances of a more general notation known as
2152:
5952:
5093:
4682:
4560:
4247:
4235:
961:
1190:
5566:
5445:
4979:
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4231:
4168:
1481:
2786:
1337:{\displaystyle \{x\mid x\in E{\text{ and }}\Phi (x)\}\quad {\text{or}}\quad \{x\mid x\in E\land \Phi (x)\}.}
792:
5809:
4998:
4579:
4306:
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313:
1643:
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5259:
4243:
225:
2745:
1112:
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365:
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2600:
is a notational variant for the same set of even natural numbers. It is not necessary to specify that
5491:
5376:
5364:
5359:
2476:
1033:
719:
164:
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for which the predicate is true. This can easily lead to contradictions and paradoxes. For example,
629:
539:
5292:
3114:
2739:
871:
597:
179:
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3002:
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5656:
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creates a set of pairs, where each pair puts an integer into correspondence with an odd integer.
2708:{\displaystyle \{a\in \mathbb {R} \mid (\exists p\in \mathbb {Z} )(\exists q\in \mathbb {Z} )\}}
772:
5863:
5744:
5556:
5369:
5109:
4271:
4022:
3887:{\displaystyle (x\in \mathbb {R} \land x^{2}=1)\Leftrightarrow (x\in \mathbb {Q} \land |x|=1).}
3341:
2314:
5130:
3977:
1861:
1394:
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4227:
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127:
5208:
3306:
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1604:
1354:
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5677:
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5501:
5381:
4021:, set builder notation is not part of the formal syntax of the theory. Instead, there is a
3941:
1637:
844:
904:
characterization may be done informally using general prose, as in the following example.
8:
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910:
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5506:
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4221:
1103:
for which the predicate holds (is true) belong to the set being defined. All values of
577:
667:
303:
When it is desired to denote a set that contains elements from a regular sequence, an
5628:
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5428:
5398:
5322:
5234:
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1765:
170:
131:
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4251:
119:
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4165:
obtained from this axiom is exactly the set described in set builder notation as
3783:{\displaystyle \{x\in \mathbb {R} \mid x^{2}=1\}=\{x\in \mathbb {Q} \mid |x|=1\}}
3482:{\displaystyle \{2t+1\mid t\in \mathbb {Z} \}=\{u\mid (u-1)/2\in \mathbb {Z} \}.}
2716:
1758:
2942:{\displaystyle \{f(x)\mid \Phi (x)\}=\{y\mid \exists x(y=f(x)\wedge \Phi (x))\}}
5921:
5718:
5699:
5603:
5588:
5545:
5481:
5423:
2442:
2291:{\displaystyle \{(x,y)\in \mathbb {R} \times \mathbb {R} \mid 0<y<f(x)\}}
1348:
407:
5941:
5926:
5728:
5642:
5637:
2053:{\displaystyle G_{m}=\{x\in \mathbb {Z} \mid x\geq m\}=\{m,m+1,m+2,\ldots \}}
972:
otherwise. In this form, set-builder notation has three parts: a variable, a
135:
5896:
4312:
Consider these set-builder notation examples in some programming languages:
2727:
5876:
5871:
5689:
5618:
5576:
5435:
5332:
5097:
977:
96:
3021:
is the set of all natural numbers, is the set of all even natural numbers.
220:
is the set containing the four numbers 3, 7, 15, and 31, and nothing else.
5901:
5536:
5183:
2439:
2142:{\displaystyle G_{3}=\{x\in \mathbb {Z} \mid x\geq 3\}=\{3,4,5,\ldots \}}
1761:
115:
960:
Set-builder notation can be used to describe a set that is defined by a
5881:
5652:
5308:
521:{\displaystyle \{\ldots ,-2,-1,0,1,2,\ldots \}=\{0,1,-1,2,-2,\ldots \}}
107:
1624:
In general, it is not a good idea to consider sets without defining a
1109:
for which the predicate does not hold do not belong to the set. Thus
5684:
5647:
5598:
5496:
1161:
767:
151:
2606:
is a natural number, as this is implied by the formula on the right.
304:
300:
This is sometimes called the "roster method" for specifying a set.
296:, and nothing else (there is no order among the elements of a set).
2732:
An extension of set-builder notation replaces the single variable
2720:
4039:
is a formula in the language of set theory, then there is a set
2719:; that is, real numbers that can be written as the ratio of two
2469:. The ∃ sign stands for "there exists", which is known as
1077:
The vertical bar (or colon) is a separator that can be read as "
5709:
5531:
4846:
1629:
5155:
Richard
Aufmann, Vernon C. Barker, and Joanne Lockwood, 2007,
5581:
5341:
5277:
5135:(6th ed.). New York, NY: McGraw-Hill. pp. 111–112.
111:
4455:{\displaystyle \{(k,x)\ |\ k\in K\wedge x\in X\wedge P(x)\}}
1947:
1081:", "for which", or "with the property that". The formula
138:, or stating the properties that its members must satisfy.
3793:
because the two rule predicates are logically equivalent:
4776:
3492:
2728:
More complex expressions on the left side of the notation
307:
notation may be employed, as shown in the next examples:
5233:
Irvine, Andrew David; Deutsch, Harry (9 October 2016) .
3082:{\displaystyle \{p/q\mid p,q\in \mathbb {Z} ,q\not =0\}}
1594:{\displaystyle \{x\in E\mid \Phi _{1}(x),\Phi _{2}(x)\}}
84:{\displaystyle \{n\mid \exists k\in \mathbb {Z} ,n=2k\}}
3574:{\displaystyle \{x\in A\mid P(x)\}=\{x\in B\mid Q(x)\}}
4383:
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2593:{\displaystyle \{n\mid (\exists k\in \mathbb {N} )\}}
2539:
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2431:{\displaystyle \{n\in \mathbb {N} \mid (\exists k)\}}
2363:
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1965:
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418:
368:
316:
228:
182:
39:
5092:, which permits map/filter-like operations over any
3897:
This equivalence holds because, for any real number
1375:. This notation represents the set of all values of
1371:
symbol denotes the logical "and" operator, known as
360:
is the set of integers between 1 and 100 inclusive.
4454:
4368:
4204:
4149:{\displaystyle (\forall E)(\exists Y)(\forall x).}
4148:
4001:
3966:
3926:
3886:
3782:
3677:
3573:
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3243:{\displaystyle \{(t,2t+1)\mid t\in \mathbb {Z} \}}
3242:
3179:
3128:
3103:
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3013:
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618:
586:
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398:
352:
276:
212:
83:
4270:It has been suggested that parts of this page be
2355:denotes the set of ordered pairs of real numbers.
1939:{\displaystyle \{x\in \mathbb {R} \mid x^{2}=1\}}
5939:
3974:. In particular, both sets are equal to the set
1750:{\displaystyle \{x\in \mathbb {R} \mid x>0\}}
2348:{\displaystyle \mathbb {R} \times \mathbb {R} }
1851:{\displaystyle \{x\in \mathbb {R} \mid |x|=1\}}
964:, that is, a logical formula that evaluates to
3296:{\displaystyle \{2t+1\mid t\in \mathbb {Z} \}}
3180:{\displaystyle \{2t+1\mid t\in \mathbb {Z} \}}
2298:is the set of pairs of real numbers such that
1471:{\displaystyle \Phi _{1}(x)\land \Phi _{2}(x)}
955:
158:
5293:
5232:
1184:can appear on the left of the vertical bar:
141:Defining sets by properties is also known as
4449:
4384:
4363:
4337:
4226:A similar notation available in a number of
4199:
4172:
3996:
3981:
3777:
3741:
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3196:
3174:
3145:
3076:
3030:
2992:{\displaystyle \{2n\mid n\in \mathbb {N} \}}
2986:
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2770:
2749:
2702:
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2011:
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467:
419:
393:
369:
347:
317:
271:
253:
247:
229:
207:
183:
78:
40:
4215:
947:is the set of all addresses on Pine Street.
5300:
5286:
4045:whose members are exactly the elements of
2208:{\displaystyle G_{-2}=\{-2,-1,0,\ldots \}}
5132:Discrete Mathematics and its Applications
3852:
3813:
3751:
3712:
3469:
3423:
3286:
3233:
3170:
3119:
3097:
3060:
3007:
2982:
2665:
2645:
2625:
2562:
2465:sign stands for "and", which is known as
2403:
2374:
2341:
2333:
2254:
2246:
2090:
1989:
1910:
1819:
1728:
59:
1388:
1230:{\displaystyle \{x\in E\mid \Phi (x)\},}
1173:
165:Set (mathematics) § Roster notation
5170:A Transition to Mathematics with Proofs
4205:{\displaystyle \{x\in E\mid \Phi (x)\}}
1518:{\displaystyle \{x\in E\mid \Phi (x)\}}
574:denotes the set of all natural numbers
14:
5940:
5172:, Jones & Bartlett, pp. 44ff.
5157:Intermediate Algebra with Applications
4012:
3493:Equivalent predicates yield equal sets
2829:{\displaystyle \{f(x)\mid \Phi (x)\},}
1948:equivalent predicates yield equal sets
1601:, using a comma instead of the symbol
1387:for which the predicate is true (see "
834:{\displaystyle \{a_{1},\dots ,a_{n}\}}
5281:
5181:
5128:
4017:In many formal set theories, such as
3388:in the set builder notation to find
1240:or by adjoining it to the predicate:
353:{\displaystyle \{1,2,3,\ldots ,100\}}
4256:
1682:{\displaystyle \{x~|~x\not \in x\},}
5239:Stanford Encyclopedia of Philosophy
277:{\displaystyle \{a,c,b\}=\{a,b,c\}}
24:
4187:
4128:
4092:
4080:
4068:
3597:
2921:
2891:
2864:
2808:
2776:{\displaystyle \{x\mid \Phi (x)\}}
2758:
2655:
2635:
2552:
2483:
2384:
1893:. This set can also be defined as
1784:
1634:all possible things that may exist
1570:
1548:
1500:
1450:
1428:
1398:
1316:
1273:
1209:
1143:{\displaystyle \{x\mid \Phi (x)\}}
1125:
1046:
1017:{\displaystyle \{x\mid \Phi (x)\}}
999:
776:
101:expressed in set-builder notation.
49:
25:
5964:
399:{\displaystyle \{1,2,3,\ldots \}}
27:Use of braces for specifying sets
5331:
4369:{\displaystyle \{l\ |\ l\in L\}}
4261:
3136:the set of all rational numbers.
2302:is greater than 0 and less than
1347:The ∈ symbol here denotes
3111:is the set of all integers, is
2507:{\displaystyle (\exists x)P(x)}
1294:
1288:
1067:{\displaystyle \{x:\Phi (x)\}.}
968:for an element of the set, and
759:{\displaystyle =\{1,\dots ,0\}}
5307:
5252:
5226:
5201:
5175:
5162:
5149:
5122:
4446:
4440:
4406:
4399:
4387:
4347:
4196:
4190:
4140:
4137:
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4098:
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3759:
3672:
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3639:
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3633:
3627:
3609:
3606:
3603:
3594:
3565:
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3532:
3526:
3454:
3442:
3363:
3351:
3220:
3199:
2933:
2930:
2924:
2915:
2909:
2897:
2873:
2867:
2858:
2852:
2817:
2811:
2802:
2796:
2767:
2761:
2699:
2672:
2669:
2652:
2649:
2632:
2584:
2569:
2566:
2549:
2501:
2495:
2489:
2480:
2422:
2393:
2390:
2381:
2282:
2276:
2239:
2227:
1835:
1827:
1787:
1775:
1657:
1628:, as this would represent the
1585:
1579:
1563:
1557:
1509:
1503:
1465:
1459:
1443:
1437:
1407:
1401:
1381:that belong to some given set
1325:
1319:
1282:
1276:
1218:
1212:
1134:
1128:
1055:
1049:
1008:
1002:
729:
723:
677:
671:
657:{\displaystyle \{1,\dots ,n\}}
567:{\displaystyle \{1,\dots ,n\}}
13:
1:
3129:{\displaystyle \mathbb {Q} ,}
2514:is read as "there exists an
893:{\displaystyle 1\leq i\leq n}
619:{\displaystyle 1\leq i\leq n}
213:{\displaystyle \{7,3,15,31\}}
4305:The same can be achieved in
4250:operations over one or more
3104:{\displaystyle \mathbb {Z} }
3014:{\displaystyle \mathbb {N} }
1150:is the set of all values of
7:
5103:
4019:Zermelo–Fraenkel set theory
3678:{\displaystyle (\forall t)}
3187:is the set of odd integers.
1793:{\displaystyle (0,\infty )}
1757:is the set of all strictly
1702:
956:Sets defined by a predicate
690:. A subtle special case is
528:is the set of all integers.
159:Sets defined by enumeration
10:
5969:
5798:von Neumann–Bernays–Gödel
5168:Michael J Cullinan, 2012,
4219:
4023:set existence axiom scheme
3938:is a rational number with
2471:existential quantification
1764:, which can be written in
1640:shows that the expression
782:{\displaystyle \emptyset }
162:
5862:
5825:
5737:
5627:
5599:One-to-one correspondence
5515:
5456:
5340:
5329:
5315:
5260:"Sequence Comprehensions"
5159:, Brooks Cole, p. 6.
4845:
4775:
4681:
4578:
4559:
4465:
3377:{\displaystyle t=(u-1)/2}
1156:that satisfy the formula
927:addresses on Pine Street
5115:
5049:
5022:
5009:
5003:
4990:
4984:
4893:
4851:
4799:
4781:
4711:
4687:
4614:
4584:
4571:
4565:
4498:
4471:
4216:In programming languages
4002:{\displaystyle \{-1,1\}}
3303:. Make the substitution
1886:{\displaystyle \{-1,1\}}
1413:{\displaystyle \Phi (x)}
664:is the bracket notation
110:and its applications to
5129:Rosen, Kenneth (2007).
4025:, which states that if
3927:{\displaystyle x^{2}=1}
1692:In cases where the set
1170:satisfies the formula.
841:denotes the set of all
626:. Another notation for
149:or as defining a set's
5557:Constructible universe
5377:Constructibility (V=L)
5209:"Set-Builder Notation"
5110:Glossary of set theory
4456:
4370:
4206:
4150:
4003:
3968:
3928:
3888:
3784:
3679:
3575:
3483:
3378:
3332:
3331:{\displaystyle u=2t+1}
3297:
3244:
3181:
3130:
3105:
3083:
3015:
2993:
2943:
2830:
2777:
2709:
2594:
2508:
2459:
2458:{\displaystyle \land }
2432:
2349:
2292:
2209:
2143:
2054:
1940:
1887:
1852:
1794:
1751:
1683:
1615:
1614:{\displaystyle \land }
1595:
1519:
1472:
1414:
1365:
1364:{\displaystyle \land }
1338:
1231:
1144:
1068:
1018:
941:
921:
894:
862:
835:
783:
760:
710:
684:
658:
620:
588:
568:
522:
400:
354:
284:is the set containing
278:
214:
92:
85:
5953:Mathematical notation
5780:Principia Mathematica
5614:Transfinite induction
5473:(i.e. set difference)
5188:mathworld.wolfram.com
4457:
4371:
4228:programming languages
4207:
4151:
4004:
3969:
3967:{\displaystyle |x|=1}
3929:
3889:
3785:
3680:
3576:
3484:
3379:
3333:
3298:
3245:
3182:
3131:
3106:
3084:
3016:
2994:
2944:
2836:which should be read
2831:
2778:
2710:
2595:
2509:
2460:
2433:
2350:
2293:
2210:
2144:
2055:
1941:
1888:
1853:
1795:
1752:
1684:
1616:
1596:
1525:is sometimes written
1520:
1473:
1415:
1366:
1339:
1232:
1174:Specifying the domain
1145:
1069:
1019:
942:
922:
895:
863:
861:{\displaystyle a_{i}}
836:
784:
761:
711:
685:
659:
621:
589:
569:
523:
401:
355:
279:
215:
128:mathematical notation
86:
32:
5854:Burali-Forti paradox
5609:Set-builder notation
5562:Continuum hypothesis
5502:Symmetric difference
5090:monad comprehensions
4381:
4334:
4169:
4062:
3978:
3942:
3905:
3800:
3699:
3591:
3505:
3395:
3342:
3307:
3258:
3193:
3142:
3115:
3093:
3027:
3003:
2960:
2843:
2787:
2746:
2612:
2537:
2477:
2449:
2361:
2329:
2221:
2153:
2064:
1963:
1897:
1862:
1806:
1772:
1715:
1644:
1605:
1529:
1482:
1424:
1395:
1355:
1247:
1191:
1113:
1034:
987:
931:
911:
872:
845:
793:
773:
720:
694:
668:
630:
598:
578:
540:
416:
366:
314:
226:
180:
124:set-builder notation
37:
18:Set builder notation
5815:Tarski–Grothendieck
5235:"Russell's Paradox"
5182:Weisstein, Eric W.
4013:Set existence axiom
2467:logical conjunction
1626:domain of discourse
1389:Set existence axiom
1373:logical conjunction
709:{\displaystyle n=0}
134:by enumerating its
5404:Limitation of size
4452:
4366:
4276:List comprehension
4240:list comprehension
4222:List comprehension
4202:
4146:
3999:
3964:
3924:
3884:
3780:
3675:
3571:
3479:
3374:
3338:, which is to say
3328:
3293:
3240:
3177:
3126:
3101:
3079:
3011:
2989:
2939:
2826:
2773:
2705:
2590:
2504:
2473:. So for example,
2455:
2438:is the set of all
2428:
2345:
2288:
2205:
2139:
2050:
1936:
1883:
1848:
1790:
1747:
1679:
1611:
1591:
1515:
1468:
1410:
1361:
1334:
1227:
1140:
1089:is said to be the
1064:
1014:
940:{\displaystyle \}}
937:
920:{\displaystyle \{}
917:
890:
858:
831:
779:
756:
706:
680:
654:
616:
584:
564:
518:
396:
350:
274:
210:
81:
5935:
5934:
5844:Russell's paradox
5793:Zermelo–Fraenkel
5694:Dedekind-infinite
5567:Diagonal argument
5466:Cartesian product
5323:Set (mathematics)
5142:978-0-07-288008-3
5086:
5085:
4412:
4404:
4353:
4345:
4296:
4295:
4242:, which combines
2742:. So instead of
2324:cartesian product
2060:. As an example,
1953:For each integer
1766:interval notation
1663:
1655:
1638:Russell's paradox
1420:is a conjunction
1292:
1271:
1164:, if no value of
587:{\displaystyle i}
143:set comprehension
130:for describing a
16:(Redirected from
5960:
5917:Bertrand Russell
5907:John von Neumann
5892:Abraham Fraenkel
5887:Richard Dedekind
5849:Suslin's problem
5760:Cantor's theorem
5477:De Morgan's laws
5335:
5302:
5295:
5288:
5279:
5278:
5272:
5271:
5269:
5267:
5256:
5250:
5249:
5247:
5245:
5230:
5224:
5223:
5221:
5219:
5205:
5199:
5198:
5196:
5194:
5179:
5173:
5166:
5160:
5153:
5147:
5146:
5126:
5080:
5077:
5074:
5071:
5068:
5065:
5062:
5059:
5056:
5053:
5044:
5041:
5038:
5035:
5032:
5029:
5026:
4972:
4969:
4966:
4963:
4960:
4957:
4954:
4951:
4948:
4945:
4942:
4939:
4936:
4933:
4930:
4927:
4924:
4921:
4918:
4915:
4912:
4909:
4906:
4903:
4900:
4897:
4888:
4885:
4882:
4879:
4876:
4873:
4870:
4867:
4864:
4861:
4858:
4855:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4794:
4791:
4788:
4785:
4769:
4766:
4763:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4739:
4736:
4733:
4730:
4727:
4724:
4721:
4718:
4715:
4706:
4703:
4700:
4697:
4694:
4691:
4675:
4672:
4669:
4666:
4663:
4660:
4657:
4654:
4651:
4648:
4645:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4609:
4606:
4603:
4600:
4597:
4594:
4591:
4588:
4553:
4550:
4547:
4544:
4541:
4538:
4535:
4532:
4529:
4526:
4523:
4520:
4517:
4514:
4511:
4508:
4505:
4502:
4493:
4490:
4487:
4484:
4481:
4478:
4475:
4461:
4459:
4458:
4453:
4410:
4409:
4402:
4375:
4373:
4372:
4367:
4351:
4350:
4343:
4315:
4314:
4291:
4288:
4265:
4264:
4257:
4211:
4209:
4208:
4203:
4164:
4155:
4153:
4152:
4147:
4054:
4050:
4044:
4038:
4030:
4008:
4006:
4005:
4000:
3973:
3971:
3970:
3965:
3957:
3949:
3933:
3931:
3930:
3925:
3917:
3916:
3893:
3891:
3890:
3885:
3871:
3863:
3855:
3829:
3828:
3816:
3789:
3787:
3786:
3781:
3770:
3762:
3754:
3728:
3727:
3715:
3684:
3682:
3681:
3676:
3584:if and only if
3580:
3578:
3577:
3572:
3488:
3486:
3485:
3480:
3472:
3461:
3426:
3383:
3381:
3380:
3375:
3370:
3337:
3335:
3334:
3329:
3302:
3300:
3299:
3294:
3289:
3249:
3247:
3246:
3241:
3236:
3186:
3184:
3183:
3178:
3173:
3135:
3133:
3132:
3127:
3122:
3110:
3108:
3107:
3102:
3100:
3088:
3086:
3085:
3080:
3063:
3040:
3020:
3018:
3017:
3012:
3010:
2998:
2996:
2995:
2990:
2985:
2948:
2946:
2945:
2940:
2835:
2833:
2832:
2827:
2782:
2780:
2779:
2774:
2737:
2717:rational numbers
2714:
2712:
2711:
2706:
2668:
2648:
2628:
2605:
2599:
2597:
2596:
2591:
2565:
2530:
2519:
2513:
2511:
2510:
2505:
2464:
2462:
2461:
2456:
2437:
2435:
2434:
2429:
2406:
2377:
2354:
2352:
2351:
2346:
2344:
2336:
2321:
2312:
2297:
2295:
2294:
2289:
2257:
2249:
2214:
2212:
2211:
2206:
2168:
2167:
2148:
2146:
2145:
2140:
2093:
2076:
2075:
2059:
2057:
2056:
2051:
1992:
1975:
1974:
1959:, we can define
1958:
1945:
1943:
1942:
1937:
1926:
1925:
1913:
1892:
1890:
1889:
1884:
1857:
1855:
1854:
1849:
1838:
1830:
1822:
1799:
1797:
1796:
1791:
1756:
1754:
1753:
1748:
1731:
1697:
1688:
1686:
1685:
1680:
1661:
1660:
1653:
1620:
1618:
1617:
1612:
1600:
1598:
1597:
1592:
1578:
1577:
1556:
1555:
1524:
1522:
1521:
1516:
1477:
1475:
1474:
1469:
1458:
1457:
1436:
1435:
1419:
1417:
1416:
1411:
1386:
1380:
1370:
1368:
1367:
1362:
1343:
1341:
1340:
1335:
1293:
1290:
1272:
1269:
1236:
1234:
1233:
1228:
1183:
1169:
1160:. It may be the
1159:
1155:
1149:
1147:
1146:
1141:
1108:
1102:
1097:. All values of
1088:
1073:
1071:
1070:
1065:
1023:
1021:
1020:
1015:
946:
944:
943:
938:
926:
924:
923:
918:
899:
897:
896:
891:
867:
865:
864:
859:
857:
856:
840:
838:
837:
832:
827:
826:
808:
807:
788:
786:
785:
780:
766:is equal to the
765:
763:
762:
757:
715:
713:
712:
707:
689:
687:
686:
683:{\displaystyle }
681:
663:
661:
660:
655:
625:
623:
622:
617:
593:
591:
590:
585:
573:
571:
570:
565:
527:
525:
524:
519:
405:
403:
402:
397:
359:
357:
356:
351:
295:
291:
287:
283:
281:
280:
275:
219:
217:
216:
211:
120:computer science
102:
90:
88:
87:
82:
62:
21:
5968:
5967:
5963:
5962:
5961:
5959:
5958:
5957:
5938:
5937:
5936:
5931:
5858:
5837:
5821:
5786:New Foundations
5733:
5623:
5542:Cardinal number
5525:
5511:
5452:
5336:
5327:
5311:
5306:
5276:
5275:
5265:
5263:
5258:
5257:
5253:
5243:
5241:
5231:
5227:
5217:
5215:
5207:
5206:
5202:
5192:
5190:
5180:
5176:
5167:
5163:
5154:
5150:
5143:
5127:
5123:
5118:
5106:
5082:
5081:
5078:
5075:
5072:
5069:
5066:
5063:
5060:
5057:
5054:
5051:
5046:
5045:
5042:
5039:
5036:
5033:
5030:
5027:
5024:
5012:
5011:
5006:
5005:
4993:
4992:
4987:
4986:
4974:
4973:
4970:
4967:
4964:
4961:
4958:
4955:
4952:
4949:
4946:
4943:
4940:
4937:
4934:
4931:
4928:
4925:
4922:
4919:
4916:
4913:
4910:
4907:
4904:
4901:
4898:
4895:
4890:
4889:
4886:
4883:
4880:
4877:
4874:
4871:
4868:
4865:
4862:
4859:
4856:
4853:
4841:
4840:
4837:
4834:
4831:
4828:
4825:
4822:
4819:
4816:
4813:
4810:
4807:
4804:
4801:
4796:
4795:
4792:
4789:
4786:
4783:
4771:
4770:
4767:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4708:
4707:
4704:
4701:
4698:
4695:
4692:
4689:
4677:
4676:
4673:
4670:
4667:
4664:
4661:
4658:
4655:
4652:
4649:
4646:
4643:
4640:
4637:
4634:
4631:
4628:
4625:
4622:
4619:
4616:
4611:
4610:
4607:
4604:
4601:
4598:
4595:
4592:
4589:
4586:
4574:
4573:
4568:
4567:
4555:
4554:
4551:
4548:
4545:
4542:
4539:
4536:
4533:
4530:
4527:
4524:
4521:
4518:
4515:
4512:
4509:
4506:
4503:
4500:
4495:
4494:
4491:
4488:
4485:
4482:
4479:
4476:
4473:
4405:
4382:
4379:
4378:
4346:
4335:
4332:
4331:
4292:
4286:
4283:
4266:
4262:
4224:
4218:
4170:
4167:
4166:
4160:
4063:
4060:
4059:
4052:
4046:
4040:
4032:
4026:
4015:
3979:
3976:
3975:
3953:
3945:
3943:
3940:
3939:
3934:if and only if
3912:
3908:
3906:
3903:
3902:
3867:
3859:
3851:
3824:
3820:
3812:
3801:
3798:
3797:
3766:
3758:
3750:
3723:
3719:
3711:
3700:
3697:
3696:
3592:
3589:
3588:
3506:
3503:
3502:
3495:
3468:
3457:
3422:
3396:
3393:
3392:
3384:, then replace
3366:
3343:
3340:
3339:
3308:
3305:
3304:
3285:
3259:
3256:
3255:
3232:
3194:
3191:
3190:
3169:
3143:
3140:
3139:
3118:
3116:
3113:
3112:
3096:
3094:
3091:
3090:
3059:
3036:
3028:
3025:
3024:
3006:
3004:
3001:
3000:
2981:
2961:
2958:
2957:
2844:
2841:
2840:
2788:
2785:
2784:
2747:
2744:
2743:
2733:
2730:
2664:
2644:
2624:
2613:
2610:
2609:
2601:
2561:
2538:
2535:
2534:
2521:
2515:
2478:
2475:
2474:
2450:
2447:
2446:
2443:natural numbers
2402:
2373:
2362:
2359:
2358:
2340:
2332:
2330:
2327:
2326:
2317:
2303:
2253:
2245:
2222:
2219:
2218:
2160:
2156:
2154:
2151:
2150:
2089:
2071:
2067:
2065:
2062:
2061:
1988:
1970:
1966:
1964:
1961:
1960:
1954:
1921:
1917:
1909:
1898:
1895:
1894:
1863:
1860:
1859:
1834:
1826:
1818:
1807:
1804:
1803:
1773:
1770:
1769:
1727:
1716:
1713:
1712:
1705:
1693:
1656:
1645:
1642:
1641:
1606:
1603:
1602:
1573:
1569:
1551:
1547:
1530:
1527:
1526:
1483:
1480:
1479:
1453:
1449:
1431:
1427:
1425:
1422:
1421:
1396:
1393:
1392:
1382:
1376:
1356:
1353:
1352:
1289:
1270: and
1268:
1248:
1245:
1244:
1192:
1189:
1188:
1179:
1176:
1165:
1157:
1151:
1114:
1111:
1110:
1104:
1098:
1082:
1035:
1032:
1031:
988:
985:
984:
958:
932:
929:
928:
912:
909:
908:
873:
870:
869:
852:
848:
846:
843:
842:
822:
818:
803:
799:
794:
791:
790:
774:
771:
770:
721:
718:
717:
695:
692:
691:
669:
666:
665:
631:
628:
627:
599:
596:
595:
579:
576:
575:
541:
538:
537:
417:
414:
413:
408:natural numbers
367:
364:
363:
315:
312:
311:
293:
289:
285:
227:
224:
223:
181:
178:
177:
167:
161:
147:set abstraction
104:
100:
95:The set of all
94:
58:
38:
35:
34:
28:
23:
22:
15:
12:
11:
5:
5966:
5956:
5955:
5950:
5933:
5932:
5930:
5929:
5924:
5922:Thoralf Skolem
5919:
5914:
5909:
5904:
5899:
5894:
5889:
5884:
5879:
5874:
5868:
5866:
5860:
5859:
5857:
5856:
5851:
5846:
5840:
5838:
5836:
5835:
5832:
5826:
5823:
5822:
5820:
5819:
5818:
5817:
5812:
5807:
5806:
5805:
5790:
5789:
5788:
5776:
5775:
5774:
5763:
5762:
5757:
5752:
5747:
5741:
5739:
5735:
5734:
5732:
5731:
5726:
5721:
5716:
5707:
5702:
5697:
5687:
5682:
5681:
5680:
5675:
5670:
5660:
5650:
5645:
5640:
5634:
5632:
5625:
5624:
5622:
5621:
5616:
5611:
5606:
5604:Ordinal number
5601:
5596:
5591:
5586:
5585:
5584:
5579:
5569:
5564:
5559:
5554:
5549:
5539:
5534:
5528:
5526:
5524:
5523:
5520:
5516:
5513:
5512:
5510:
5509:
5504:
5499:
5494:
5489:
5484:
5482:Disjoint union
5479:
5474:
5468:
5462:
5460:
5454:
5453:
5451:
5450:
5449:
5448:
5443:
5432:
5431:
5429:Martin's axiom
5426:
5421:
5416:
5411:
5406:
5401:
5396:
5394:Extensionality
5391:
5390:
5389:
5379:
5374:
5373:
5372:
5367:
5362:
5352:
5346:
5344:
5338:
5337:
5330:
5328:
5326:
5325:
5319:
5317:
5313:
5312:
5305:
5304:
5297:
5290:
5282:
5274:
5273:
5251:
5225:
5213:mathsisfun.com
5200:
5174:
5161:
5148:
5141:
5120:
5119:
5117:
5114:
5113:
5112:
5105:
5102:
5084:
5083:
5050:
5047:
5023:
5020:
5014:
5013:
5010:
5007:
5004:
5001:
4995:
4994:
4991:
4988:
4985:
4982:
4976:
4975:
4894:
4891:
4852:
4849:
4843:
4842:
4800:
4797:
4782:
4779:
4773:
4772:
4712:
4709:
4688:
4685:
4679:
4678:
4615:
4612:
4585:
4582:
4576:
4575:
4572:
4569:
4566:
4563:
4557:
4556:
4499:
4496:
4472:
4469:
4463:
4462:
4451:
4448:
4445:
4442:
4439:
4436:
4433:
4430:
4427:
4424:
4421:
4418:
4415:
4408:
4401:
4398:
4395:
4392:
4389:
4386:
4376:
4365:
4362:
4359:
4356:
4349:
4342:
4339:
4329:
4325:
4324:
4321:
4318:
4294:
4293:
4269:
4267:
4260:
4220:Main article:
4217:
4214:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4174:
4157:
4156:
4145:
4142:
4139:
4136:
4133:
4130:
4127:
4124:
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4014:
4011:
3998:
3995:
3992:
3989:
3986:
3983:
3963:
3960:
3956:
3952:
3948:
3923:
3920:
3915:
3911:
3895:
3894:
3883:
3880:
3877:
3874:
3870:
3866:
3862:
3858:
3854:
3850:
3847:
3844:
3841:
3838:
3835:
3832:
3827:
3823:
3819:
3815:
3811:
3808:
3805:
3791:
3790:
3779:
3776:
3773:
3769:
3765:
3761:
3757:
3753:
3749:
3746:
3743:
3740:
3737:
3734:
3731:
3726:
3722:
3718:
3714:
3710:
3707:
3704:
3687:
3686:
3674:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3582:
3581:
3570:
3567:
3564:
3561:
3558:
3555:
3552:
3549:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3494:
3491:
3490:
3489:
3478:
3475:
3471:
3467:
3464:
3460:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3425:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3373:
3369:
3365:
3362:
3359:
3356:
3353:
3350:
3347:
3327:
3324:
3321:
3318:
3315:
3312:
3292:
3288:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3252:
3251:
3239:
3235:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3188:
3176:
3172:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3137:
3125:
3121:
3099:
3078:
3075:
3072:
3069:
3066:
3062:
3058:
3055:
3052:
3049:
3046:
3043:
3039:
3035:
3032:
3022:
3009:
2988:
2984:
2980:
2977:
2974:
2971:
2968:
2965:
2951:
2950:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2783:, we may have
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2729:
2726:
2725:
2724:
2715:is the set of
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2667:
2663:
2660:
2657:
2654:
2651:
2647:
2643:
2640:
2637:
2634:
2631:
2627:
2623:
2620:
2617:
2607:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2564:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2532:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2454:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2405:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2376:
2372:
2369:
2366:
2356:
2343:
2339:
2335:
2313:, for a given
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2256:
2252:
2248:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2216:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2166:
2163:
2159:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2092:
2088:
2085:
2082:
2079:
2074:
2070:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1991:
1987:
1984:
1981:
1978:
1973:
1969:
1951:
1935:
1932:
1929:
1924:
1920:
1916:
1912:
1908:
1905:
1902:
1882:
1879:
1876:
1873:
1870:
1867:
1847:
1844:
1841:
1837:
1833:
1829:
1825:
1821:
1817:
1814:
1811:
1801:
1789:
1786:
1783:
1780:
1777:
1746:
1743:
1740:
1737:
1734:
1730:
1726:
1723:
1720:
1704:
1701:
1678:
1675:
1672:
1669:
1666:
1659:
1652:
1649:
1610:
1590:
1587:
1584:
1581:
1576:
1572:
1568:
1565:
1562:
1559:
1554:
1550:
1546:
1543:
1540:
1537:
1534:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1467:
1464:
1461:
1456:
1452:
1448:
1445:
1442:
1439:
1434:
1430:
1409:
1406:
1403:
1400:
1360:
1349:set membership
1345:
1344:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1287:
1284:
1281:
1278:
1275:
1267:
1264:
1261:
1258:
1255:
1252:
1238:
1237:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1175:
1172:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1075:
1074:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1025:
1024:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
957:
954:
949:
948:
936:
916:
889:
886:
883:
880:
877:
855:
851:
830:
825:
821:
817:
814:
811:
806:
802:
798:
778:
755:
752:
749:
746:
743:
740:
737:
734:
731:
728:
725:
705:
702:
699:
679:
676:
673:
653:
650:
647:
644:
641:
638:
635:
615:
612:
609:
606:
603:
583:
563:
560:
557:
554:
551:
548:
545:
530:
529:
517:
514:
511:
508:
505:
502:
499:
496:
493:
490:
487:
484:
481:
478:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
411:
406:is the set of
395:
392:
389:
386:
383:
380:
377:
374:
371:
361:
349:
346:
343:
340:
337:
334:
331:
328:
325:
322:
319:
298:
297:
273:
270:
267:
264:
261:
258:
255:
252:
249:
246:
243:
240:
237:
234:
231:
221:
209:
206:
203:
200:
197:
194:
191:
188:
185:
163:Main article:
160:
157:
80:
77:
74:
71:
68:
65:
61:
57:
54:
51:
48:
45:
42:
31:
26:
9:
6:
4:
3:
2:
5965:
5954:
5951:
5949:
5946:
5945:
5943:
5928:
5927:Ernst Zermelo
5925:
5923:
5920:
5918:
5915:
5913:
5912:Willard Quine
5910:
5908:
5905:
5903:
5900:
5898:
5895:
5893:
5890:
5888:
5885:
5883:
5880:
5878:
5875:
5873:
5870:
5869:
5867:
5865:
5864:Set theorists
5861:
5855:
5852:
5850:
5847:
5845:
5842:
5841:
5839:
5833:
5831:
5828:
5827:
5824:
5816:
5813:
5811:
5810:Kripke–Platek
5808:
5804:
5801:
5800:
5799:
5796:
5795:
5794:
5791:
5787:
5784:
5783:
5782:
5781:
5777:
5773:
5770:
5769:
5768:
5765:
5764:
5761:
5758:
5756:
5753:
5751:
5748:
5746:
5743:
5742:
5740:
5736:
5730:
5727:
5725:
5722:
5720:
5717:
5715:
5713:
5708:
5706:
5703:
5701:
5698:
5695:
5691:
5688:
5686:
5683:
5679:
5676:
5674:
5671:
5669:
5666:
5665:
5664:
5661:
5658:
5654:
5651:
5649:
5646:
5644:
5641:
5639:
5636:
5635:
5633:
5630:
5626:
5620:
5617:
5615:
5612:
5610:
5607:
5605:
5602:
5600:
5597:
5595:
5592:
5590:
5587:
5583:
5580:
5578:
5575:
5574:
5573:
5570:
5568:
5565:
5563:
5560:
5558:
5555:
5553:
5550:
5547:
5543:
5540:
5538:
5535:
5533:
5530:
5529:
5527:
5521:
5518:
5517:
5514:
5508:
5505:
5503:
5500:
5498:
5495:
5493:
5490:
5488:
5485:
5483:
5480:
5478:
5475:
5472:
5469:
5467:
5464:
5463:
5461:
5459:
5455:
5447:
5446:specification
5444:
5442:
5439:
5438:
5437:
5434:
5433:
5430:
5427:
5425:
5422:
5420:
5417:
5415:
5412:
5410:
5407:
5405:
5402:
5400:
5397:
5395:
5392:
5388:
5385:
5384:
5383:
5380:
5378:
5375:
5371:
5368:
5366:
5363:
5361:
5358:
5357:
5356:
5353:
5351:
5348:
5347:
5345:
5343:
5339:
5334:
5324:
5321:
5320:
5318:
5314:
5310:
5303:
5298:
5296:
5291:
5289:
5284:
5283:
5280:
5261:
5255:
5240:
5236:
5229:
5214:
5210:
5204:
5189:
5185:
5178:
5171:
5165:
5158:
5152:
5144:
5138:
5134:
5133:
5125:
5121:
5111:
5108:
5107:
5101:
5099:
5095:
5091:
5048:
5021:
5019:
5016:
5015:
5008:
5002:
5000:
4997:
4996:
4989:
4983:
4981:
4978:
4977:
4892:
4850:
4848:
4844:
4798:
4780:
4778:
4774:
4710:
4686:
4684:
4680:
4613:
4583:
4581:
4577:
4570:
4564:
4562:
4558:
4497:
4470:
4468:
4464:
4443:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4413:
4396:
4393:
4390:
4377:
4360:
4357:
4354:
4340:
4330:
4327:
4326:
4322:
4319:
4317:
4316:
4313:
4310:
4308:
4303:
4301:
4290:
4287:December 2023
4281:
4277:
4273:
4268:
4259:
4258:
4255:
4253:
4249:
4245:
4241:
4237:
4233:
4229:
4223:
4213:
4193:
4184:
4181:
4178:
4175:
4163:
4143:
4134:
4125:
4122:
4119:
4116:
4110:
4107:
4104:
4095:
4083:
4071:
4058:
4057:
4056:
4051:that satisfy
4049:
4043:
4036:
4031:is a set and
4029:
4024:
4020:
4010:
3993:
3990:
3987:
3984:
3961:
3958:
3950:
3937:
3921:
3918:
3913:
3909:
3900:
3881:
3875:
3872:
3864:
3856:
3848:
3845:
3833:
3830:
3825:
3821:
3817:
3809:
3806:
3796:
3795:
3794:
3774:
3771:
3763:
3755:
3747:
3744:
3738:
3732:
3729:
3724:
3720:
3716:
3708:
3705:
3695:
3694:
3693:
3692:For example,
3690:
3663:
3657:
3654:
3651:
3648:
3645:
3630:
3624:
3621:
3618:
3615:
3612:
3600:
3587:
3586:
3585:
3562:
3556:
3553:
3550:
3547:
3544:
3538:
3529:
3523:
3520:
3517:
3514:
3511:
3501:
3500:
3499:
3476:
3465:
3462:
3458:
3451:
3448:
3445:
3439:
3436:
3430:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3391:
3390:
3389:
3387:
3371:
3367:
3360:
3357:
3354:
3348:
3345:
3325:
3322:
3319:
3316:
3313:
3310:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3229:
3226:
3223:
3217:
3214:
3211:
3208:
3205:
3202:
3189:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3138:
3123:
3073:
3070:
3067:
3064:
3056:
3053:
3050:
3047:
3044:
3041:
3037:
3033:
3023:
2978:
2975:
2972:
2969:
2966:
2956:
2955:
2954:
2953:For example:
2927:
2918:
2912:
2906:
2903:
2900:
2894:
2888:
2885:
2879:
2870:
2861:
2855:
2849:
2839:
2838:
2837:
2823:
2814:
2805:
2799:
2793:
2764:
2755:
2752:
2741:
2736:
2722:
2718:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2661:
2658:
2641:
2638:
2629:
2621:
2618:
2608:
2604:
2581:
2578:
2575:
2572:
2558:
2555:
2546:
2543:
2533:
2528:
2524:
2518:
2498:
2492:
2486:
2472:
2468:
2452:
2444:
2441:
2419:
2416:
2413:
2410:
2407:
2399:
2396:
2387:
2378:
2370:
2367:
2357:
2337:
2325:
2322:. Here the
2320:
2316:
2310:
2306:
2301:
2279:
2273:
2270:
2267:
2264:
2261:
2258:
2250:
2242:
2236:
2233:
2230:
2217:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2169:
2164:
2161:
2157:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2109:
2103:
2100:
2097:
2094:
2086:
2083:
2077:
2072:
2068:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2008:
2002:
1999:
1996:
1993:
1985:
1982:
1976:
1971:
1967:
1957:
1952:
1949:
1930:
1927:
1922:
1918:
1914:
1906:
1903:
1877:
1874:
1871:
1868:
1842:
1839:
1831:
1823:
1815:
1812:
1802:
1781:
1778:
1767:
1763:
1760:
1741:
1738:
1735:
1732:
1724:
1721:
1711:
1710:
1709:
1700:
1696:
1690:
1676:
1670:
1667:
1664:
1650:
1639:
1635:
1631:
1627:
1622:
1608:
1582:
1574:
1566:
1560:
1552:
1544:
1541:
1538:
1535:
1506:
1497:
1494:
1491:
1488:
1462:
1454:
1446:
1440:
1432:
1404:
1391:" below). If
1390:
1385:
1379:
1374:
1358:
1350:
1331:
1322:
1313:
1310:
1307:
1304:
1301:
1298:
1279:
1265:
1262:
1259:
1256:
1253:
1243:
1242:
1241:
1224:
1215:
1206:
1203:
1200:
1197:
1187:
1186:
1185:
1182:
1171:
1168:
1163:
1154:
1131:
1122:
1119:
1107:
1101:
1096:
1092:
1086:
1080:
1061:
1052:
1043:
1040:
1030:
1029:
1028:
1005:
996:
993:
983:
982:
981:
979:
975:
971:
967:
963:
953:
907:
906:
905:
901:
887:
884:
881:
878:
875:
853:
849:
823:
819:
815:
812:
809:
804:
800:
789:. Similarly,
769:
750:
747:
744:
741:
738:
732:
726:
703:
700:
697:
674:
648:
645:
642:
639:
636:
613:
610:
607:
604:
601:
581:
558:
555:
552:
549:
546:
534:
512:
509:
506:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
470:
464:
461:
458:
455:
452:
449:
446:
443:
440:
437:
434:
431:
428:
425:
422:
412:
409:
390:
387:
384:
381:
378:
375:
372:
362:
344:
341:
338:
335:
332:
329:
326:
323:
320:
310:
309:
308:
306:
301:
268:
265:
262:
259:
256:
250:
244:
241:
238:
235:
232:
222:
204:
201:
198:
195:
192:
189:
186:
176:
175:
174:
172:
166:
156:
154:
153:
148:
144:
139:
137:
133:
129:
125:
121:
117:
113:
109:
103:
98:
97:even integers
91:
75:
72:
69:
66:
63:
55:
52:
46:
43:
30:
19:
5877:Georg Cantor
5872:Paul Bernays
5803:Morse–Kelley
5778:
5711:
5710:Subset
5657:hereditarily
5619:Venn diagram
5608:
5577:ordered pair
5492:Intersection
5436:Axiom schema
5264:. Retrieved
5254:
5242:. Retrieved
5238:
5228:
5216:. Retrieved
5212:
5203:
5191:. Retrieved
5187:
5177:
5169:
5164:
5156:
5151:
5131:
5124:
5098:zero element
5089:
5087:
4328:Set-builder
4311:
4304:
4297:
4284:
4225:
4161:
4158:
4047:
4041:
4034:
4027:
4016:
3935:
3898:
3896:
3792:
3691:
3688:
3583:
3496:
3385:
3253:
2952:
2734:
2731:
2602:
2526:
2522:
2516:
2318:
2308:
2304:
2299:
1955:
1762:real numbers
1706:
1694:
1691:
1633:
1623:
1383:
1377:
1351:, while the
1346:
1239:
1180:
1177:
1166:
1152:
1105:
1099:
1094:
1090:
1084:
1078:
1076:
1026:
978:vertical bar
969:
965:
959:
950:
902:
536:In general,
535:
531:
302:
299:
168:
150:
146:
142:
140:
123:
105:
93:
33:
29:
5902:Thomas Jech
5745:Alternative
5724:Uncountable
5678:Ultrafilter
5537:Cardinality
5441:replacement
5382:Determinacy
5018:Mathematica
1858:is the set
716:, in which
116:mathematics
5948:Set theory
5942:Categories
5897:Kurt Gödel
5882:Paul Cohen
5719:Transitive
5487:Identities
5471:Complement
5458:Operations
5419:Regularity
5387:projective
5350:Adjunction
5309:Set theory
4323:Example 2
3901:, we have
2740:expression
2520:such that
594:such that
108:set theory
5830:Paradoxes
5750:Axiomatic
5729:Universal
5705:Singleton
5700:Recursive
5643:Countable
5638:Amorphous
5497:Power set
5414:Power set
5365:dependent
5360:countable
5218:20 August
5193:20 August
4435:∧
4429:∈
4423:∧
4417:∈
4358:∈
4320:Example 1
4300:generator
4238:) is the
4230:(notably
4188:Φ
4185:∣
4179:∈
4129:Φ
4126:∧
4120:∈
4114:⇔
4108:∈
4093:∀
4081:∃
4069:∀
3985:−
3857:∧
3849:∈
3840:⇔
3818:∧
3810:∈
3756:∣
3748:∈
3717:∣
3709:∈
3655:∧
3649:∈
3640:⇔
3622:∧
3616:∈
3598:∀
3554:∣
3548:∈
3521:∣
3515:∈
3466:∈
3449:−
3440:∣
3420:∈
3414:∣
3358:−
3283:∈
3277:∣
3230:∈
3224:∣
3167:∈
3161:∣
3057:∈
3045:∣
2979:∈
2973:∣
2922:Φ
2919:∧
2892:∃
2889:∣
2865:Φ
2862:∣
2809:Φ
2806:∣
2759:Φ
2756:∣
2685:∧
2662:∈
2656:∃
2642:∈
2636:∃
2630:∣
2622:∈
2559:∈
2553:∃
2547:∣
2484:∃
2453:∧
2408:∧
2400:∈
2385:∃
2379:∣
2371:∈
2338:×
2259:∣
2251:×
2243:∈
2200:…
2185:−
2176:−
2162:−
2134:…
2101:≥
2095:∣
2087:∈
2045:…
2000:≥
1994:∣
1986:∈
1915:∣
1907:∈
1869:−
1824:∣
1816:∈
1785:∞
1733:∣
1725:∈
1609:∧
1571:Φ
1549:Φ
1545:∣
1539:∈
1501:Φ
1498:∣
1492:∈
1451:Φ
1447:∧
1429:Φ
1399:Φ
1359:∧
1317:Φ
1314:∧
1308:∈
1302:∣
1274:Φ
1263:∈
1257:∣
1210:Φ
1207:∣
1201:∈
1178:A domain
1162:empty set
1126:Φ
1123:∣
1095:predicate
1079:such that
1047:Φ
1000:Φ
997:∣
962:predicate
885:≤
879:≤
813:…
777:∅
768:empty set
745:…
643:…
611:≤
605:≤
553:…
513:…
504:−
489:−
465:…
438:−
429:−
423:…
391:…
339:…
152:intension
56:∈
50:∃
47:∣
5834:Problems
5738:Theories
5714:Superset
5690:Infinite
5519:Concepts
5399:Infinity
5316:Overview
5266:6 August
5244:6 August
5104:See also
4159:The set
3089:, where
3071:≠
2999:, where
2738:with an
2721:integers
2679:≠
2315:function
1759:positive
1703:Examples
1668:∉
305:ellipsis
136:elements
5772:General
5767:Zermelo
5673:subbase
5655: (
5594:Forcing
5572:Element
5544: (
5522:Methods
5409:Pairing
5262:. Scala
5096:with a
4561:Haskell
4280:Discuss
4236:Haskell
1478:, then
1093:or the
1083:Φ(
5663:Filter
5653:Finite
5589:Family
5532:Almost
5370:global
5355:Choice
5342:Axioms
5139:
5031:|->
4980:Erlang
4968:Result
4932:member
4914:member
4884:Result
4866:member
4847:Prolog
4802:SELECT
4784:SELECT
4753:select
4702:select
4467:Python
4411:
4403:
4352:
4344:
4248:filter
4232:Python
2445:. The
1950:below.
1946:; see
1662:
1654:
1630:subset
1158:Φ
292:, and
118:, and
5755:Naive
5685:Fuzzy
5648:Empty
5631:types
5582:tuple
5552:Class
5546:large
5507:Union
5424:Union
5184:"Set"
5116:Notes
5094:monad
5052:Cases
4999:Julia
4896:setof
4854:setof
4826:WHERE
4823:X_set
4817:K_set
4793:L_set
4738:where
4659:yield
4638:<-
4626:<-
4605:yield
4596:<-
4580:Scala
4307:Scala
4274:into
4272:moved
4252:lists
974:colon
970:false
126:is a
112:logic
5668:base
5268:2017
5246:2017
5220:2020
5195:2020
5137:ISBN
4950:call
4814:FROM
4790:FROM
4726:from
4714:from
4690:from
4246:and
4234:and
2440:even
2271:<
2265:<
2149:and
1739:>
1091:rule
966:true
868:for
5629:Set
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4911:),(
4777:SQL
4617:for
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4278:. (
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1768:as
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1027:or
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5944::
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742:,
739:1
736:{
733:=
730:]
727:0
724:[
704:0
701:=
698:n
678:]
675:n
672:[
652:}
649:n
646:,
640:,
637:1
634:{
614:n
608:i
602:1
582:i
562:}
559:n
556:,
550:,
547:1
544:{
516:}
510:,
507:2
501:,
498:2
495:,
492:1
486:,
483:1
480:,
477:0
474:{
471:=
468:}
462:,
459:2
456:,
453:1
450:,
447:0
444:,
441:1
435:,
432:2
426:,
420:{
410:.
394:}
388:,
385:3
382:,
379:2
376:,
373:1
370:{
348:}
342:,
336:,
333:3
330:,
327:2
324:,
321:1
318:{
294:c
290:b
286:a
272:}
269:c
266:,
263:b
260:,
257:a
254:{
251:=
248:}
245:b
242:,
239:c
236:,
233:a
230:{
208:}
202:,
196:,
193:3
190:,
187:7
184:{
79:}
76:k
73:2
70:=
67:n
64:,
60:Z
53:k
44:n
41:{
20:)
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