8085:
774:
5744:
40:
1731:
1482:
1635:
683:
2458:
2594:
2874:, and where each pair makes up one element of the output set. The number of values in each element of the resulting set is equal to the number of sets whose Cartesian product is being taken; 2 in this case. The cardinality of the output set is equal to the product of the cardinalities of all the input sets. That is,
3728:
2238:
2797:
3187:
4083:
3419:
2462:
668:
3554:
4564:
2706:
2699:
3038:
1965:
3933:
3265:
212:
2453:{\displaystyle {\begin{aligned}A\times (B\cap C)&=(A\times B)\cap (A\times C),\\A\times (B\cup C)&=(A\times B)\cup (A\times C),\\A\times (B\setminus C)&=(A\times B)\setminus (A\times C),\end{aligned}}}
1873:
540:
1422:
4303:
1037:
467:
2231:
2119:
4446:
2243:
837:
4827:
4628:
1201:
4504:
2589:{\displaystyle (A\times B)^{\complement }=\left(A^{\complement }\times B^{\complement }\right)\cup \left(A^{\complement }\times B\right)\cup \left(A\times B^{\complement }\right)\!,}
953:
4597:
5114:
4155:
3928:
3878:
415:
4354:
4220:
2621:
6464:
1087:
491:
4179:
5066:
4887:
4484:
2644:
859:
1121:, and relations are usually defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is necessarily prior to most other definitions.
4957:
1063:
7139:
6201:
3549:
985:
754:
returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), ..., (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}.
365:
5086:
5044:
5020:
4997:
4977:
4931:
4911:
4861:
535:
515:
329:
309:
1884:
7222:
6363:
130:
3723:{\displaystyle X^{n}=\underbrace {X\times X\times \cdots \times X} _{n}=\{(x_{1},\ldots ,x_{n})\ |\ x_{i}\in X\ {\text{for every}}\ i\in \{1,\ldots ,n\}\}.}
1792:
870:
7536:
1365:
2792:{\displaystyle {\text{if both }}A,B\neq \emptyset {\text{, then }}A\times B\subseteq C\times D\!\iff \!A\subseteq C{\text{ and }}B\subseteq D.}
7694:
4236:
6482:
2126:
1972:
7549:
6872:
5890:
5710:
4386:
1471:
789:. In order to represent geometrical shapes in a numerical way, and extract numerical information from shapes' numerical representations,
7134:
6218:
7554:
7544:
7281:
6487:
5472:
5138:
3182:{\displaystyle X_{1}\times \cdots \times X_{n}=\{(x_{1},\ldots ,x_{n})\mid x_{i}\in X_{i}\ {\text{for every}}\ i\in \{1,\ldots ,n\}\}}
33:
7032:
6478:
7690:
5564:
4746:
7787:
7531:
6356:
4078:{\displaystyle \prod _{i\in I}X_{i}=\left\{\left.f:I\to \bigcup _{i\in I}X_{i}\ \right|\ \forall i\in I.\ f(i)\in X_{i}\right\},}
3414:{\displaystyle \{x:\{1,\ldots ,n\}\to X_{1}\cup \cdots \cup X_{n}\ |\ x(i)\in X_{i}\ {\text{for every}}\ i\in \{1,\ldots ,n\}\}.}
7092:
6785:
6526:
6196:
4835:
and infinite collections of functions. This is different from the standard
Cartesian product of functions considered as sets.
1165:
8048:
7750:
7513:
7508:
7333:
6754:
6438:
990:
420:
291:. In this case the domain would have to contain the Cartesian product itself. For defining the Cartesian product of the sets
8043:
7826:
7743:
7456:
7387:
7264:
6506:
5970:
5849:
5676:
1110:
7114:
6213:
7968:
7794:
7480:
6713:
7119:
6206:
216:
A table can be created by taking the
Cartesian product of a set of rows and a set of columns. If the Cartesian product
8114:
7451:
7190:
6448:
6349:
5844:
5807:
5671:
5176:
3507:
663:{\displaystyle A\times B=\{x\in {\mathcal {P}}({\mathcal {P}}(A\cup B))\mid \exists a\in A\ \exists b\in B:x=(a,b)\}.}
7846:
7841:
812:
7775:
7365:
6759:
6727:
6418:
5603:
6492:
8065:
8014:
7911:
7409:
7370:
6847:
5895:
5787:
5775:
5770:
5355:
5303:
5160:
5134:
4602:
1786:
7906:
6521:
8109:
7836:
7375:
7227:
7210:
6933:
6413:
5703:
5307:
4362:
890:
7738:
7715:
7676:
7562:
7503:
7149:
7069:
6913:
6857:
6470:
6315:
6233:
6108:
6060:
5874:
5797:
4573:
4461:
8028:
7755:
7733:
7700:
7593:
7439:
7424:
7397:
7348:
7232:
7167:
6992:
6958:
6953:
6827:
6658:
6635:
6267:
6148:
5960:
5780:
5683:
5091:
5023:
4117:
1118:
1114:
710:} form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52
7958:
7811:
7603:
7321:
7057:
6963:
6822:
6807:
6688:
6663:
6183:
6097:
6017:
5997:
5975:
5197:
3887:
3837:
691:
8084:
370:
4308:
4183:
2599:
7931:
7893:
7770:
7574:
7414:
7338:
7316:
7144:
7102:
7001:
6968:
6832:
6620:
6531:
6257:
6247:
6081:
6012:
5965:
5905:
5792:
288:
124:
5666:
1068:
472:
8060:
7951:
7936:
7916:
7873:
7760:
7710:
7636:
7581:
7518:
7311:
7306:
7254:
7022:
7011:
6683:
6583:
6511:
6502:
6498:
6433:
6428:
6252:
6163:
6076:
6071:
6066:
5880:
5822:
5760:
5696:
4160:
3454:
2807:
2624:
5049:
4866:
4467:
842:
8089:
7858:
7821:
7806:
7799:
7782:
7568:
7434:
7360:
7343:
7296:
7109:
7018:
6852:
6837:
6797:
6749:
6734:
6722:
6678:
6653:
6423:
6372:
6175:
6170:
5955:
5910:
5817:
5634:, p. 24. Studies in Logic and the Foundations of Mathematics, vol. 76 (1978). ISBN 0-7204-2200-0.
5427:
5319:
5148:
1878:
1106:
7586:
7042:
4559:{\displaystyle \prod _{n=1}^{\infty }\mathbb {R} =\mathbb {R} \times \mathbb {R} \times \cdots }
809:
coordinates, respectively (see picture). The set of all such pairs (i.e., the
Cartesian product
4936:
1042:
8024:
7831:
7641:
7631:
7523:
7404:
7239:
7215:
6996:
6980:
6885:
6739:
6708:
6673:
6568:
6403:
6032:
5861:
5832:
5802:
5733:
5560:
5529:
5391:
5335:
2235:
Here are some rules demonstrating distributivity with other operators (see leftmost picture):
790:
786:
773:
272:
268:
72:
5554:
8038:
8033:
7926:
7883:
7705:
7666:
7661:
7646:
7472:
7429:
7326:
7124:
7074:
6648:
6610:
6320:
6310:
6295:
6290:
6158:
5812:
5360:
4567:
4114:, which is equivalent to the statement that every such product is nonempty, is not assumed.
1098:
5480:
3527:
958:
338:
8019:
8009:
7963:
7946:
7901:
7863:
7765:
7685:
7492:
7419:
7392:
7380:
7286:
7200:
7174:
7129:
7097:
6898:
6700:
6643:
6593:
6558:
6516:
6189:
6127:
5945:
5765:
5130:
4111:
782:
5137:
of mathematical structures. This is distinct from, although related to, the notion of a
287:
A rigorous definition of the
Cartesian product requires a domain to be specified in the
8004:
7983:
7941:
7921:
7816:
7671:
7269:
7259:
7249:
7244:
7178:
7052:
6928:
6817:
6812:
6790:
6391:
6325:
6122:
6103:
6007:
5992:
5949:
5885:
5827:
5502:
5442:
5325:
5071:
5029:
5005:
4982:
4962:
4916:
4896:
4846:
4487:
3823:
3813:
3794:
1102:
520:
500:
314:
294:
276:
261:
39:
5394:
8103:
7978:
7656:
7163:
6948:
6938:
6908:
6893:
6563:
6330:
6300:
6132:
6046:
6041:
5350:
5330:
5152:
5142:
3790:
2694:{\displaystyle {\text{if }}A\subseteq B{\text{, then }}A\times C\subseteq B\times C;}
758:
698:
ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13-element set. The card suits
695:
7878:
7725:
7626:
7618:
7498:
7446:
7355:
7291:
7274:
7205:
7064:
6923:
6625:
6408:
6280:
6275:
6093:
6022:
5980:
5839:
5743:
5660:
5365:
5172:
5156:
2979:
1204:
884:
880:
861:
denoting the real numbers) is thus assigned to the set of all points in the plane.
711:
332:
93:
5306:
in the sense of category theory. Instead, the categorical product is known as the
32:"Cartesian square" redirects here. For Cartesian squares in category theory, see
7988:
7868:
7047:
7037:
6984:
6668:
6588:
6573:
6453:
6398:
6305:
5940:
5619:
5604:
https://proofwiki.org/search/?title=Cartesian_Product_of_Subsets&oldid=45868
5345:
3774:
3477:
2813:
1359:
1159:
798:
794:
60:
4570:
with countably infinite real number components. This set is frequently denoted
2816:
of a set is the number of elements of the set. For example, defining two sets:
1877:
In most cases, the above statement is not true if we replace intersection with
1093:
operator. Therefore, the existence of the
Cartesian product of any two sets in
17:
6918:
6773:
6744:
6550:
6285:
6153:
6056:
5719:
5340:
876:
64:
8070:
7973:
7026:
6943:
6903:
6867:
6803:
6615:
6605:
6578:
6088:
6051:
6002:
5900:
5399:
5155:
of the
Cartesian product; thus any category with a Cartesian product (and a
4086:
3831:
1229:
1090:
765:
between them, under which (3, ♣) corresponds to (♣, 3) and so on.
762:
494:
1730:
1481:
1207:
are reversed unless at least one of the following conditions is satisfied:
4667:), then some authors choose to abbreviate the Cartesian product as simply
1634:
682:
8055:
7853:
7301:
7006:
6600:
3819:
3818:
It is possible to define the
Cartesian product of an arbitrary (possibly
3798:
1960:{\displaystyle (A\cup B)\times (C\cup D)\neq (A\times C)\cup (B\times D)}
7651:
6443:
4490:: this Cartesian product is the set of all infinite sequences with the
1785:
The
Cartesian product satisfies the following property with respect to
207:{\displaystyle A\times B=\{(a,b)\mid a\in A\ {\mbox{ and }}\ b\in B\}.}
6341:
6113:
5935:
5614:
Peter S. (1998). A Crash Course in the
Mathematics of Infinite Sets.
5587:
2635:
1868:{\displaystyle (A\cap B)\times (C\cap D)=(A\times C)\cap (B\times D)}
734:), (A, ♣), (K, ♠), ..., (3, ♣), (2, ♠), (2,
801:. Usually, such a pair's first and second components are called its
275:
gave rise to the concept, which is further generalized in terms of
260:. More generally still, one can define the Cartesian product of an
226:
is taken, the cells of the table contain ordered pairs of the form
7195:
6541:
6386:
5985:
5752:
4832:
3199:
3192:
2838:
consist of two elements each. Their
Cartesian product, written as
772:
681:
257:
253:
38:
5129:
Although the Cartesian product is traditionally applied to sets,
6345:
5692:
5559:. Dover Books on Mathematics. Courier Corporation. p. 41.
1094:
5688:
2123:
For the set difference, we also have the following identity:
1074:
1006:
996:
577:
567:
478:
436:
426:
5684:
How to find the Cartesian Product, Education Portal Academy
3972:
1417:{\displaystyle (A\times B)\times C\neq A\times (B\times C)}
4092:
such that the value of the function at a particular index
2848:, results in a new set which has the following elements:
5346:
Orders on the Cartesian product of totally ordered sets
4298:{\displaystyle \pi _{j}:\prod _{i\in I}X_{i}\to X_{j},}
4110:
is nonempty, the Cartesian product may be empty if the
1032:{\displaystyle {\mathcal {P}}({\mathcal {P}}(X\cup Y))}
462:{\displaystyle {\mathcal {P}}({\mathcal {P}}(A\cup B))}
2226:{\displaystyle (A\times C)\setminus (B\times D)=\cup }
2114:{\displaystyle (A\times C)\cup (B\times D)=\cup \cup }
180:
5141:
in category theory, which is a generalization of the
5094:
5074:
5052:
5032:
5008:
4985:
4965:
4939:
4919:
4899:
4869:
4849:
4749:
4643:
If several sets are being multiplied together (e.g.,
4605:
4576:
4507:
4470:
4464:. An important special case is when the index set is
4441:{\displaystyle \prod _{i\in I}X_{i}=\prod _{i\in I}X}
4389:
4311:
4239:
4186:
4163:
4120:
3936:
3890:
3840:
3557:
3530:
3268:
3244:-th element of the tuple, then the Cartesian product
3041:
2990:
is infinite, and the other set is not the empty set.
2709:
2647:
2602:
2465:
2241:
2129:
1975:
1887:
1795:
1368:
1168:
1071:
1045:
993:
961:
893:
845:
815:
714:, which correspond to all 52 possible playing cards.
543:
523:
503:
475:
423:
373:
341:
317:
297:
133:
7997:
7892:
7724:
7617:
7469:
7162:
7085:
6979:
6883:
6772:
6699:
6634:
6549:
6540:
6462:
6379:
6266:
6229:
6141:
6031:
5919:
5860:
5751:
5726:
5598:Cartesian Product of Subsets. (February 15, 2011).
5108:
5080:
5060:
5038:
5014:
4991:
4971:
4951:
4925:
4905:
4881:
4855:
4821:
4622:
4591:
4558:
4478:
4440:
4348:
4297:
4214:
4173:
4149:
4077:
3922:
3872:
3785:. As a special case, the 0-ary Cartesian power of
3756:again the set of real numbers, and more generally
3722:
3543:
3413:
3181:
2791:
2693:
2615:
2588:
2452:
2225:
2113:
1959:
1867:
1416:
1195:
1081:
1057:
1031:
979:
947:
875:A formal definition of the Cartesian product from
853:
831:
662:
529:
509:
485:
461:
409:
359:
323:
303:
233:One can similarly define the Cartesian product of
206:
5322:(to prove the existence of the Cartesian product)
5026:of the context and is left away. For example, if
4085:that is, the set of all functions defined on the
2762:
2756:
2582:
726:returns a set of the form {(A, ♠), (A,
3006:The Cartesian product can be generalized to the
1358:Strictly speaking, the Cartesian product is not
883:. The most common definition of ordered pairs,
832:{\displaystyle \mathbb {R} \times \mathbb {R} }
5632:Set Theory: An Introduction to Large Cardinals
5133:provides a more general interpretation of the
1311:= {1,2} × {1,2} = {(1,1), (1,2), (2,1), (2,2)}
1275:= {3,4} × {1,2} = {(3,1), (3,2), (4,1), (4,2)}
1263:= {1,2} × {3,4} = {(1,3), (1,4), (2,3), (2,4)}
793:assigned to each point in the plane a pair of
6357:
5704:
5620:http://www.mathpath.org/concepts/infinity.htm
4822:{\displaystyle (f\times g)(x,y)=(f(x),g(y)).}
4372:is a Cartesian product where all the factors
248:-dimensional array, where each element is an
8:
3905:
3891:
3884:, then the Cartesian product of the sets in
3855:
3841:
3714:
3711:
3693:
3611:
3405:
3402:
3384:
3296:
3278:
3269:
3176:
3173:
3155:
3074:
1362:(unless one of the involved sets is empty).
942:
939:
927:
921:
915:
912:
654:
556:
404:
401:
389:
383:
377:
374:
198:
146:
5618:(2), 35–59. Retrieved August 1, 2011, from
5302:. The Cartesian product of graphs is not a
4623:{\displaystyle \mathbb {R} ^{\mathbb {N} }}
871:Implementation of mathematics in set theory
27:Mathematical set formed from two given sets
7183:
6778:
6546:
6364:
6350:
6342:
5711:
5697:
5689:
5586:. Retrieved from the Connexions Web site:
2761:
2757:
5530:"A Sketch of the Rudiments of Set Theory"
5102:
5101:
5093:
5073:
5054:
5053:
5051:
5031:
5007:
4984:
4964:
4938:
4918:
4898:
4868:
4848:
4748:
4614:
4613:
4612:
4608:
4607:
4604:
4583:
4579:
4578:
4575:
4546:
4545:
4538:
4537:
4530:
4529:
4523:
4512:
4506:
4472:
4471:
4469:
4460:. This case is important in the study of
4423:
4410:
4394:
4388:
4316:
4310:
4286:
4273:
4257:
4244:
4238:
4206:
4190:
4188:
4185:
4165:
4164:
4162:
4141:
4125:
4119:
4061:
4007:
3991:
3957:
3941:
3935:
3908:
3898:
3889:
3858:
3848:
3839:
3679:
3664:
3652:
3640:
3621:
3602:
3572:
3562:
3556:
3535:
3529:
3370:
3361:
3334:
3325:
3306:
3267:
3229:. If a tuple is defined as a function on
3141:
3132:
3119:
3103:
3084:
3065:
3046:
3040:
2772:
2730:
2710:
2708:
2662:
2648:
2646:
2607:
2601:
2571:
2536:
2513:
2500:
2482:
2464:
2242:
2240:
2128:
1974:
1886:
1794:
1367:
1167:
1117:are usually defined as a special case of
1073:
1072:
1070:
1044:
1005:
1004:
995:
994:
992:
960:
892:
847:
846:
844:
825:
824:
817:
816:
814:
576:
575:
566:
565:
542:
522:
502:
497:. Then the Cartesian product of the sets
477:
476:
474:
435:
434:
425:
424:
422:
372:
340:
316:
296:
179:
132:
5200:set is the (ordinary) Cartesian product
1196:{\displaystyle A\times B\neq B\times A,}
879:principles follows from a definition of
5643:Osborne, M., and Rubinstein, A., 1994.
5602:. Retrieved 05:06, August 1, 2011 from
5377:
2425:
2394:
2205:
2181:
2145:
2093:
2021:
1125:Non-commutativity and non-associativity
948:{\displaystyle (x,y)=\{\{x\},\{x,y\}\}}
865:Most common implementation (set theory)
777:Cartesian coordinates of example points
4592:{\displaystyle \mathbb {R} ^{\omega }}
4166:
5578:
5576:
267:The Cartesian product is named after
7:
5467:
5465:
5463:
5385:
5383:
5381:
5109:{\displaystyle B\times \mathbb {N} }
4150:{\displaystyle \prod _{i\in I}X_{i}}
1472:List of set identities and relations
5046:is a subset of the natural numbers
781:The main historical example is the
769:A two-dimensional coordinate system
417:, an appropriate domain is the set
5588:http://cnx.org/content/m15207/1.5/
4524:
4494:-th term in its corresponding set
4024:
3923:{\displaystyle \{X_{i}\}_{i\in I}}
3873:{\displaystyle \{X_{i}\}_{i\in I}}
3777:to the space of functions from an
3453:. An example is the 2-dimensional
2994:Cartesian products of several sets
2727:
1778:can be seen from the same example.
1466:Intersections, unions, and subsets
757:These two sets are distinct, even
618:
603:
34:Cartesian square (category theory)
25:
4448:is the set of all functions from
410:{\displaystyle \{\{a\},\{a,b\}\}}
244:, which can be represented by an
8083:
5742:
4349:{\displaystyle \pi _{j}(f)=f(j)}
4215:{\displaystyle {}_{i\in I}X_{i}}
2616:{\displaystyle A^{\complement }}
1729:
1633:
1480:
5661:Cartesian Product at ProvenMath
5177:Cartesian product of two graphs
4713:, then their Cartesian product
4501:. For example, each element of
3880:is a family of sets indexed by
2870:is paired with each element of
1065:is a subset of that set, where
690:An illustrative example is the
43:Cartesian product of the sets {
5443:"Cartesian product definition"
5120:Definitions outside set theory
4813:
4810:
4804:
4795:
4789:
4783:
4777:
4765:
4762:
4750:
4685:Cartesian product of functions
4343:
4337:
4328:
4322:
4279:
4051:
4045:
3984:
3769:-ary Cartesian power of a set
3653:
3646:
3614:
3351:
3345:
3335:
3299:
3109:
3077:
2860:= {(a,5), (a,6), (b,5), (b,6)}
2758:
2634:Other properties related with
2479:
2466:
2440:
2428:
2422:
2410:
2400:
2388:
2372:
2360:
2354:
2342:
2332:
2320:
2304:
2292:
2286:
2274:
2264:
2252:
2220:
2211:
2199:
2196:
2190:
2187:
2175:
2166:
2160:
2148:
2142:
2130:
2108:
2099:
2087:
2084:
2078:
2075:
2063:
2057:
2045:
2042:
2036:
2027:
2015:
2012:
2006:
1994:
1988:
1976:
1954:
1942:
1936:
1924:
1918:
1906:
1900:
1888:
1862:
1850:
1844:
1832:
1826:
1814:
1808:
1796:
1411:
1399:
1381:
1369:
1082:{\displaystyle {\mathcal {P}}}
1026:
1023:
1011:
1001:
974:
962:
906:
894:
651:
639:
597:
594:
582:
572:
486:{\displaystyle {\mathcal {P}}}
456:
453:
441:
431:
354:
342:
161:
149:
1:
8044:History of mathematical logic
5582:Singh, S. (August 27, 2009).
4174:{\displaystyle {\mathsf {X}}}
7969:Primitive recursive function
5556:Probability: An Introduction
5061:{\displaystyle \mathbb {N} }
4882:{\displaystyle B\subseteq A}
4479:{\displaystyle \mathbb {N} }
4456:, and is frequently denoted
3202:, it can be identified with
854:{\displaystyle \mathbb {R} }
5672:Encyclopedia of Mathematics
5219:and such that two vertices
3808:Infinite Cartesian products
3508:Cartesian coordinate system
3198:. If tuples are defined as
1097:follows from the axioms of
8131:
7033:Schröder–Bernstein theorem
6760:Monadic predicate calculus
6419:Foundations of mathematics
6202:von Neumann–Bernays–Gödel
3811:
3506:are real numbers (see the
3236:} that takes its value at
2805:
1469:
868:
31:
8079:
8066:Philosophy of mathematics
8015:Automated theorem proving
7186:
7140:Von Neumann–Bernays–Gödel
6781:
6003:One-to-one correspondence
5740:
5553:Goldberg, Samuel (1986).
5356:Product (category theory)
5161:Cartesian closed category
4952:{\displaystyle B\times A}
4933:is the Cartesian product
3486:is the set of all points
3439:is the Cartesian product
1881:(see rightmost picture).
1058:{\displaystyle X\times Y}
955:. Under this definition,
761:, but there is a natural
228:(row value, column value)
5308:tensor product of graphs
5186:is the graph denoted by
5022:is considered to be the
4831:This can be extended to
3262:is the set of functions
283:Set-theoretic definition
7716:Self-verifying theories
7537:Tarski's axiomatization
6488:Tarski's undefinability
6483:incompleteness theorems
5645:A Course in Game Theory
5341:Join (SQL) § Cross join
5068:, then the cylinder of
4566:can be visualized as a
4462:cardinal exponentiation
3793:, corresponding to the
1969:In fact, we have that:
885:Kuratowski's definition
333:Kuratowski's definition
271:, whose formulation of
256:. An ordered pair is a
242:-fold Cartesian product
237:sets, also known as an
8090:Mathematics portal
7701:Proof of impossibility
7349:propositional variable
6659:Propositional calculus
5961:Constructible universe
5788:Constructibility (V=L)
5110:
5082:
5062:
5040:
5016:
4993:
4973:
4953:
4927:
4907:
4883:
4857:
4823:
4624:
4593:
4560:
4528:
4480:
4442:
4350:
4299:
4216:
4175:
4151:
4103:. Even if each of the
4079:
3924:
3874:
3732:An example of this is
3724:
3545:
3415:
3183:
3011:-ary Cartesian product
3002:-ary Cartesian product
2866:where each element of
2793:
2695:
2617:
2590:
2454:
2227:
2115:
1961:
1869:
1789:(see middle picture).
1418:
1197:
1148:The Cartesian product
1083:
1059:
1033:
981:
949:
855:
833:
778:
687:
664:
531:
511:
487:
463:
411:
361:
325:
305:
208:
56:
7959:Kolmogorov complexity
7912:Computably enumerable
7812:Model complete theory
7604:Principia Mathematica
6664:Propositional formula
6493:Banach–Tarski paradox
6184:Principia Mathematica
6018:Transfinite induction
5877:(i.e. set difference)
5616:St. John's Review, 44
5111:
5083:
5063:
5041:
5017:
4994:
4974:
4954:
4928:
4908:
4884:
4858:
4824:
4625:
4594:
4561:
4508:
4481:
4443:
4351:
4300:
4217:
4176:
4152:
4080:
3925:
3875:
3789:may be taken to be a
3725:
3546:
3544:{\displaystyle X^{n}}
3416:
3184:
2794:
2696:
2618:
2591:
2455:
2228:
2116:
1962:
1870:
1419:
1198:
1084:
1060:
1034:
982:
980:{\displaystyle (x,y)}
950:
856:
834:
776:
696:standard playing card
692:standard 52-card deck
686:Standard 52-card deck
685:
665:
532:
512:
488:
464:
412:
362:
360:{\displaystyle (a,b)}
326:
306:
209:
42:
7907:Church–Turing thesis
7894:Computability theory
7103:continuum hypothesis
6621:Square of opposition
6479:Gödel's completeness
6258:Burali-Forti paradox
6013:Set-builder notation
5966:Continuum hypothesis
5906:Symmetric difference
5092:
5072:
5050:
5030:
5006:
4983:
4963:
4937:
4917:
4897:
4867:
4847:
4747:
4603:
4574:
4505:
4468:
4387:
4309:
4237:
4184:
4161:
4157:may also be denoted
4118:
3934:
3888:
3838:
3555:
3551:, can be defined as
3528:
3518:-ary Cartesian power
3427:-ary Cartesian power
3266:
3200:nested ordered pairs
3039:
2707:
2645:
2600:
2463:
2239:
2127:
1973:
1885:
1793:
1366:
1166:
1069:
1043:
991:
959:
891:
843:
813:
541:
537:would be defined as
521:
501:
473:
421:
371:
339:
315:
295:
289:set-builder notation
131:
125:set-builder notation
92:, is the set of all
8061:Mathematical object
7952:P versus NP problem
7917:Computable function
7711:Reverse mathematics
7637:Logical consequence
7514:primitive recursive
7509:elementary function
7282:Free/bound variable
7135:Tarski–Grothendieck
6654:Logical connectives
6584:Logical equivalence
6434:Logical consequence
6219:Tarski–Grothendieck
5507:Merriam-Webster.com
5473:"Cartesian Product"
5422:Warner, S. (1990).
5395:"Cartesian Product"
4723:is a function from
4705:is a function from
4693:is a function from
2808:Cardinal arithmetic
2625:absolute complement
331:, with the typical
8115:Operations on sets
7859:Transfer principle
7822:Semantics of logic
7807:Categorical theory
7783:Non-standard model
7297:Logical connective
6424:Information theory
6373:Mathematical logic
5808:Limitation of size
5428:Dover Publications
5392:Weisstein, Eric W.
5320:Axiom of power set
5106:
5078:
5058:
5036:
5012:
4989:
4969:
4949:
4923:
4903:
4879:
4853:
4819:
4620:
4589:
4556:
4476:
4438:
4434:
4405:
4346:
4295:
4268:
4212:
4171:
4147:
4136:
4075:
4002:
3952:
3920:
3870:
3720:
3607:
3600:
3541:
3411:
3179:
2789:
2691:
2613:
2586:
2450:
2448:
2223:
2111:
1957:
1865:
1414:
1193:
1079:
1055:
1029:
977:
945:
851:
829:
779:
688:
660:
527:
507:
483:
459:
407:
357:
321:
301:
204:
184:
57:
8097:
8096:
8029:Abstract category
7832:Theories of truth
7642:Rule of inference
7632:Natural deduction
7613:
7612:
7158:
7157:
6863:Cartesian product
6768:
6767:
6674:Many-valued logic
6649:Boolean functions
6532:Russell's paradox
6507:diagonal argument
6404:First-order logic
6339:
6338:
6248:Russell's paradox
6197:Zermelo–Fraenkel
6098:Dedekind-infinite
5971:Diagonal argument
5870:Cartesian product
5734:Set (mathematics)
5584:Cartesian product
5336:Finitary relation
5294:is adjacent with
5268:is adjacent with
5253:, if and only if
5081:{\displaystyle B}
5039:{\displaystyle B}
5015:{\displaystyle A}
4992:{\displaystyle A}
4972:{\displaystyle B}
4926:{\displaystyle A}
4906:{\displaystyle B}
4856:{\displaystyle A}
4419:
4390:
4379:are the same set
4253:
4121:
4096:is an element of
4041:
4023:
4015:
3987:
3937:
3930:is defined to be
3686:
3682:
3678:
3659:
3651:
3573:
3571:
3377:
3373:
3369:
3341:
3333:
3148:
3144:
3140:
2775:
2733:
2713:
2665:
2651:
1445:= {((1, 1), 1)} ≠
987:is an element of
787:analytic geometry
742:), (2, ♣)}.
617:
530:{\displaystyle B}
510:{\displaystyle A}
324:{\displaystyle B}
304:{\displaystyle A}
273:analytic geometry
258:2-tuple or couple
188:
183:
178:
69:Cartesian product
16:(Redirected from
8122:
8088:
8087:
8039:History of logic
8034:Category of sets
7927:Decision problem
7706:Ordinal analysis
7647:Sequent calculus
7545:Boolean algebras
7485:
7484:
7459:
7430:logical/constant
7184:
7170:
7093:Zermelo–Fraenkel
6844:Set operations:
6779:
6716:
6547:
6527:Löwenheim–Skolem
6414:Formal semantics
6366:
6359:
6352:
6343:
6321:Bertrand Russell
6311:John von Neumann
6296:Abraham Fraenkel
6291:Richard Dedekind
6253:Suslin's problem
6164:Cantor's theorem
5881:De Morgan's laws
5746:
5713:
5706:
5699:
5690:
5680:
5667:"Direct product"
5648:
5641:
5635:
5628:
5622:
5612:
5606:
5596:
5590:
5580:
5571:
5570:
5550:
5544:
5543:
5541:
5539:
5534:
5525:
5519:
5518:
5516:
5514:
5499:
5493:
5492:
5490:
5488:
5483:on July 18, 2020
5479:. Archived from
5469:
5458:
5457:
5455:
5453:
5438:
5432:
5431:
5419:
5413:
5412:
5411:
5409:
5407:
5387:
5361:Product topology
5301:
5297:
5293:
5289:
5275:
5271:
5267:
5263:
5252:
5243:are adjacent in
5242:
5230:
5218:
5195:
5185:
5181:
5139:Cartesian square
5115:
5113:
5112:
5107:
5105:
5087:
5085:
5084:
5079:
5067:
5065:
5064:
5059:
5057:
5045:
5043:
5042:
5037:
5021:
5019:
5018:
5013:
4998:
4996:
4995:
4990:
4978:
4976:
4975:
4970:
4958:
4956:
4955:
4950:
4932:
4930:
4929:
4924:
4913:with respect to
4912:
4910:
4909:
4904:
4888:
4886:
4885:
4880:
4862:
4860:
4859:
4854:
4828:
4826:
4825:
4820:
4742:
4732:
4722:
4712:
4708:
4704:
4700:
4696:
4692:
4680:
4666:
4639:Abbreviated form
4629:
4627:
4626:
4621:
4619:
4618:
4617:
4611:
4598:
4596:
4595:
4590:
4588:
4587:
4582:
4565:
4563:
4562:
4557:
4549:
4541:
4533:
4527:
4522:
4493:
4485:
4483:
4482:
4477:
4475:
4455:
4451:
4447:
4445:
4444:
4439:
4433:
4415:
4414:
4404:
4383:. In this case,
4382:
4360:
4355:
4353:
4352:
4347:
4321:
4320:
4304:
4302:
4301:
4296:
4291:
4290:
4278:
4277:
4267:
4249:
4248:
4232:
4228:
4221:
4219:
4218:
4213:
4211:
4210:
4201:
4200:
4189:
4180:
4178:
4177:
4172:
4170:
4169:
4156:
4154:
4153:
4148:
4146:
4145:
4135:
4095:
4091:
4084:
4082:
4081:
4076:
4071:
4067:
4066:
4065:
4039:
4021:
4020:
4016:
4013:
4012:
4011:
4001:
3962:
3961:
3951:
3929:
3927:
3926:
3921:
3919:
3918:
3903:
3902:
3883:
3879:
3877:
3876:
3871:
3869:
3868:
3853:
3852:
3829:
3803:
3788:
3784:
3781:-element set to
3780:
3772:
3768:
3761:
3755:
3749:
3729:
3727:
3726:
3721:
3684:
3683:
3680:
3676:
3669:
3668:
3657:
3656:
3649:
3645:
3644:
3626:
3625:
3606:
3601:
3596:
3567:
3566:
3550:
3548:
3547:
3542:
3540:
3539:
3523:
3517:
3505:
3501:
3497:
3485:
3475:
3469:
3452:
3438:
3433:Cartesian square
3420:
3418:
3417:
3412:
3375:
3374:
3371:
3367:
3366:
3365:
3339:
3338:
3331:
3330:
3329:
3311:
3310:
3261:
3243:
3239:
3235:
3228:
3195:
3188:
3186:
3185:
3180:
3146:
3145:
3142:
3138:
3137:
3136:
3124:
3123:
3108:
3107:
3089:
3088:
3070:
3069:
3051:
3050:
3034:
3016:
3010:
2989:
2985:
2977:
2961:
2959:
2953:
2947:
2941:
2920:
2918:
2902:
2900:
2894:
2888:
2873:
2869:
2861:
2847:
2837:
2833:
2829:
2822:
2798:
2796:
2795:
2790:
2776:
2773:
2734:
2731:
2714:
2711:
2700:
2698:
2697:
2692:
2666:
2663:
2652:
2649:
2630:
2622:
2620:
2619:
2614:
2612:
2611:
2595:
2593:
2592:
2587:
2581:
2577:
2576:
2575:
2552:
2548:
2541:
2540:
2523:
2519:
2518:
2517:
2505:
2504:
2487:
2486:
2459:
2457:
2456:
2451:
2449:
2232:
2230:
2229:
2224:
2120:
2118:
2117:
2112:
1966:
1964:
1963:
1958:
1874:
1872:
1871:
1866:
1777:
1775:
1765:
1755:
1733:
1723:
1721:
1711:
1701:
1678:
1676:
1668:
1666:
1659:
1657:
1650:
1648:
1637:
1628:
1626:
1616:
1591:
1589:
1579:
1555:
1553:
1543:
1533:
1516:
1514:
1506:
1504:
1497:
1495:
1484:
1461:
1448:{(1, (1, 1))} =
1446:
1430:
1423:
1421:
1420:
1415:
1352:
1340:
1328:
1312:
1292:
1276:
1264:
1252:
1245:
1227:
1223:
1217:
1213:
1202:
1200:
1199:
1194:
1157:
1144:
1140:
1136:
1132:
1088:
1086:
1085:
1080:
1078:
1077:
1064:
1062:
1061:
1056:
1038:
1036:
1035:
1030:
1010:
1009:
1000:
999:
986:
984:
983:
978:
954:
952:
951:
946:
860:
858:
857:
852:
850:
838:
836:
835:
830:
828:
820:
808:
804:
753:
741:
737:
733:
729:
725:
709:
707:
703:
669:
667:
666:
661:
615:
581:
580:
571:
570:
536:
534:
533:
528:
516:
514:
513:
508:
492:
490:
489:
484:
482:
481:
468:
466:
465:
460:
440:
439:
430:
429:
416:
414:
413:
408:
366:
364:
363:
358:
330:
328:
327:
322:
310:
308:
307:
302:
251:
247:
241:
236:
229:
225:
213:
211:
210:
205:
186:
185:
181:
176:
122:
118:
114:
110:
106:
91:
81:
77:
21:
8130:
8129:
8125:
8124:
8123:
8121:
8120:
8119:
8110:Axiom of choice
8100:
8099:
8098:
8093:
8082:
8075:
8020:Category theory
8010:Algebraic logic
7993:
7964:Lambda calculus
7902:Church encoding
7888:
7864:Truth predicate
7720:
7686:Complete theory
7609:
7478:
7474:
7470:
7465:
7457:
7177: and
7173:
7168:
7154:
7130:New Foundations
7098:axiom of choice
7081:
7043:Gödel numbering
6983: and
6975:
6879:
6764:
6714:
6695:
6644:Boolean algebra
6630:
6594:Equiconsistency
6559:Classical logic
6536:
6517:Halting problem
6505: and
6481: and
6469: and
6468:
6463:Theorems (
6458:
6375:
6370:
6340:
6335:
6262:
6241:
6225:
6190:New Foundations
6137:
6027:
5946:Cardinal number
5929:
5915:
5856:
5747:
5738:
5722:
5717:
5665:
5657:
5652:
5651:
5642:
5638:
5629:
5625:
5613:
5609:
5597:
5593:
5581:
5574:
5567:
5552:
5551:
5547:
5537:
5535:
5532:
5527:
5526:
5522:
5512:
5510:
5501:
5500:
5496:
5486:
5484:
5477:web.mnstate.edu
5471:
5470:
5461:
5451:
5449:
5441:Nykamp, Duane.
5440:
5439:
5435:
5421:
5420:
5416:
5405:
5403:
5390:
5389:
5388:
5379:
5374:
5316:
5299:
5295:
5291:
5280:
5273:
5269:
5265:
5254:
5244:
5232:
5220:
5201:
5187:
5183:
5179:
5169:
5131:category theory
5127:
5125:Category theory
5122:
5090:
5089:
5070:
5069:
5048:
5047:
5028:
5027:
5004:
5003:
4981:
4980:
4961:
4960:
4935:
4934:
4915:
4914:
4895:
4894:
4865:
4864:
4845:
4844:
4841:
4745:
4744:
4734:
4724:
4714:
4710:
4706:
4702:
4698:
4694:
4690:
4687:
4679:
4671:
4668:
4664:
4657:
4650:
4644:
4641:
4636:
4606:
4601:
4600:
4577:
4572:
4571:
4503:
4502:
4499:
4491:
4488:natural numbers
4466:
4465:
4453:
4449:
4406:
4385:
4384:
4380:
4377:
4370:Cartesian power
4358:
4312:
4307:
4306:
4282:
4269:
4240:
4235:
4234:
4233:, the function
4230:
4226:
4202:
4187:
4182:
4181:
4159:
4158:
4137:
4116:
4115:
4112:axiom of choice
4108:
4101:
4093:
4089:
4057:
4003:
3974:
3971:
3970:
3966:
3953:
3932:
3931:
3904:
3894:
3886:
3885:
3881:
3854:
3844:
3836:
3835:
3827:
3816:
3810:
3801:
3786:
3782:
3778:
3770:
3766:
3757:
3751:
3733:
3660:
3636:
3617:
3574:
3558:
3553:
3552:
3531:
3526:
3525:
3521:
3515:
3503:
3499:
3487:
3481:
3471:
3457:
3440:
3436:
3429:
3357:
3321:
3302:
3264:
3263:
3260:
3251:
3245:
3241:
3237:
3230:
3226:
3220:
3210:
3203:
3193:
3128:
3115:
3099:
3080:
3061:
3042:
3037:
3036:
3033:
3024:
3018:
3014:
3008:
3004:
2996:
2987:
2983:
2969:
2955:
2954:| · |
2949:
2948:| · |
2943:
2942:| = |
2929:
2927:
2910:
2908:
2896:
2895:| · |
2890:
2889:| = |
2880:
2878:
2871:
2867:
2852:
2839:
2835:
2831:
2824:
2817:
2810:
2804:
2774: and
2705:
2704:
2643:
2642:
2628:
2603:
2598:
2597:
2567:
2560:
2556:
2532:
2531:
2527:
2509:
2496:
2495:
2491:
2478:
2461:
2460:
2447:
2446:
2403:
2379:
2378:
2335:
2311:
2310:
2267:
2237:
2236:
2125:
2124:
1971:
1970:
1883:
1882:
1791:
1790:
1783:
1782:
1781:
1780:
1779:
1767:
1757:
1737:
1736:
1734:
1726:
1725:
1713:
1703:
1683:
1682:
1680:
1679:, demonstrating
1672:
1671:
1670:
1662:
1661:
1653:
1652:
1644:
1643:
1641:
1638:
1630:
1629:
1618:
1608:
1603: \
1595:
1593:
1581:
1571:
1558:
1557:
1545:
1535:
1520:
1519:
1518:
1517:, demonstrating
1510:
1509:
1508:
1500:
1499:
1491:
1490:
1488:
1485:
1474:
1468:
1447:
1432:
1425:
1424:If for example
1364:
1363:
1351:= ∅ × {1,2} = ∅
1343:
1339:= {1,2} × ∅ = ∅
1331:
1319:
1295:
1283:
1267:
1255:
1247:
1240:
1225:
1221:
1215:
1211:
1164:
1163:
1149:
1142:
1138:
1134:
1130:
1127:
1089:represents the
1067:
1066:
1041:
1040:
989:
988:
957:
956:
889:
888:
877:set-theoretical
873:
867:
841:
840:
811:
810:
806:
802:
783:Cartesian plane
771:
745:
739:
735:
731:
727:
717:
705:
701:
699:
680:
678:A deck of cards
675:
539:
538:
519:
518:
499:
498:
471:
470:
419:
418:
369:
368:
337:
336:
313:
312:
293:
292:
285:
249:
245:
239:
234:
227:
217:
182: and
129:
128:
120:
116:
112:
108:
96:
83:
79:
75:
63:, specifically
37:
28:
23:
22:
18:Cartesian power
15:
12:
11:
5:
8128:
8126:
8118:
8117:
8112:
8102:
8101:
8095:
8094:
8080:
8077:
8076:
8074:
8073:
8068:
8063:
8058:
8053:
8052:
8051:
8041:
8036:
8031:
8022:
8017:
8012:
8007:
8005:Abstract logic
8001:
7999:
7995:
7994:
7992:
7991:
7986:
7984:Turing machine
7981:
7976:
7971:
7966:
7961:
7956:
7955:
7954:
7949:
7944:
7939:
7934:
7924:
7922:Computable set
7919:
7914:
7909:
7904:
7898:
7896:
7890:
7889:
7887:
7886:
7881:
7876:
7871:
7866:
7861:
7856:
7851:
7850:
7849:
7844:
7839:
7829:
7824:
7819:
7817:Satisfiability
7814:
7809:
7804:
7803:
7802:
7792:
7791:
7790:
7780:
7779:
7778:
7773:
7768:
7763:
7758:
7748:
7747:
7746:
7741:
7734:Interpretation
7730:
7728:
7722:
7721:
7719:
7718:
7713:
7708:
7703:
7698:
7688:
7683:
7682:
7681:
7680:
7679:
7669:
7664:
7654:
7649:
7644:
7639:
7634:
7629:
7623:
7621:
7615:
7614:
7611:
7610:
7608:
7607:
7599:
7598:
7597:
7596:
7591:
7590:
7589:
7584:
7579:
7559:
7558:
7557:
7555:minimal axioms
7552:
7541:
7540:
7539:
7528:
7527:
7526:
7521:
7516:
7511:
7506:
7501:
7488:
7486:
7467:
7466:
7464:
7463:
7462:
7461:
7449:
7444:
7443:
7442:
7437:
7432:
7427:
7417:
7412:
7407:
7402:
7401:
7400:
7395:
7385:
7384:
7383:
7378:
7373:
7368:
7358:
7353:
7352:
7351:
7346:
7341:
7331:
7330:
7329:
7324:
7319:
7314:
7309:
7304:
7294:
7289:
7284:
7279:
7278:
7277:
7272:
7267:
7262:
7252:
7247:
7245:Formation rule
7242:
7237:
7236:
7235:
7230:
7220:
7219:
7218:
7208:
7203:
7198:
7193:
7187:
7181:
7164:Formal systems
7160:
7159:
7156:
7155:
7153:
7152:
7147:
7142:
7137:
7132:
7127:
7122:
7117:
7112:
7107:
7106:
7105:
7100:
7089:
7087:
7083:
7082:
7080:
7079:
7078:
7077:
7067:
7062:
7061:
7060:
7053:Large cardinal
7050:
7045:
7040:
7035:
7030:
7016:
7015:
7014:
7009:
7004:
6989:
6987:
6977:
6976:
6974:
6973:
6972:
6971:
6966:
6961:
6951:
6946:
6941:
6936:
6931:
6926:
6921:
6916:
6911:
6906:
6901:
6896:
6890:
6888:
6881:
6880:
6878:
6877:
6876:
6875:
6870:
6865:
6860:
6855:
6850:
6842:
6841:
6840:
6835:
6825:
6820:
6818:Extensionality
6815:
6813:Ordinal number
6810:
6800:
6795:
6794:
6793:
6782:
6776:
6770:
6769:
6766:
6765:
6763:
6762:
6757:
6752:
6747:
6742:
6737:
6732:
6731:
6730:
6720:
6719:
6718:
6705:
6703:
6697:
6696:
6694:
6693:
6692:
6691:
6686:
6681:
6671:
6666:
6661:
6656:
6651:
6646:
6640:
6638:
6632:
6631:
6629:
6628:
6623:
6618:
6613:
6608:
6603:
6598:
6597:
6596:
6586:
6581:
6576:
6571:
6566:
6561:
6555:
6553:
6544:
6538:
6537:
6535:
6534:
6529:
6524:
6519:
6514:
6509:
6497:Cantor's
6495:
6490:
6485:
6475:
6473:
6460:
6459:
6457:
6456:
6451:
6446:
6441:
6436:
6431:
6426:
6421:
6416:
6411:
6406:
6401:
6396:
6395:
6394:
6383:
6381:
6377:
6376:
6371:
6369:
6368:
6361:
6354:
6346:
6337:
6336:
6334:
6333:
6328:
6326:Thoralf Skolem
6323:
6318:
6313:
6308:
6303:
6298:
6293:
6288:
6283:
6278:
6272:
6270:
6264:
6263:
6261:
6260:
6255:
6250:
6244:
6242:
6240:
6239:
6236:
6230:
6227:
6226:
6224:
6223:
6222:
6221:
6216:
6211:
6210:
6209:
6194:
6193:
6192:
6180:
6179:
6178:
6167:
6166:
6161:
6156:
6151:
6145:
6143:
6139:
6138:
6136:
6135:
6130:
6125:
6120:
6111:
6106:
6101:
6091:
6086:
6085:
6084:
6079:
6074:
6064:
6054:
6049:
6044:
6038:
6036:
6029:
6028:
6026:
6025:
6020:
6015:
6010:
6008:Ordinal number
6005:
6000:
5995:
5990:
5989:
5988:
5983:
5973:
5968:
5963:
5958:
5953:
5943:
5938:
5932:
5930:
5928:
5927:
5924:
5920:
5917:
5916:
5914:
5913:
5908:
5903:
5898:
5893:
5888:
5886:Disjoint union
5883:
5878:
5872:
5866:
5864:
5858:
5857:
5855:
5854:
5853:
5852:
5847:
5836:
5835:
5833:Martin's axiom
5830:
5825:
5820:
5815:
5810:
5805:
5800:
5798:Extensionality
5795:
5790:
5785:
5784:
5783:
5778:
5773:
5763:
5757:
5755:
5749:
5748:
5741:
5739:
5737:
5736:
5730:
5728:
5724:
5723:
5718:
5716:
5715:
5708:
5701:
5693:
5687:
5686:
5681:
5663:
5656:
5655:External links
5653:
5650:
5649:
5636:
5623:
5607:
5591:
5572:
5565:
5545:
5520:
5494:
5459:
5433:
5424:Modern Algebra
5414:
5376:
5375:
5373:
5370:
5369:
5368:
5363:
5358:
5353:
5348:
5343:
5338:
5333:
5328:
5326:Direct product
5323:
5315:
5312:
5168:
5165:
5149:Exponentiation
5126:
5123:
5121:
5118:
5104:
5100:
5097:
5077:
5056:
5035:
5011:
4988:
4968:
4948:
4945:
4942:
4922:
4902:
4878:
4875:
4872:
4852:
4840:
4837:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4767:
4764:
4761:
4758:
4755:
4752:
4686:
4683:
4675:
4669:
4662:
4655:
4648:
4640:
4637:
4635:
4632:
4616:
4610:
4586:
4581:
4555:
4552:
4548:
4544:
4540:
4536:
4532:
4526:
4521:
4518:
4515:
4511:
4497:
4474:
4437:
4432:
4429:
4426:
4422:
4418:
4413:
4409:
4403:
4400:
4397:
4393:
4375:
4363:projection map
4356:is called the
4345:
4342:
4339:
4336:
4333:
4330:
4327:
4324:
4319:
4315:
4294:
4289:
4285:
4281:
4276:
4272:
4266:
4263:
4260:
4256:
4252:
4247:
4243:
4209:
4205:
4199:
4196:
4193:
4168:
4144:
4140:
4134:
4131:
4128:
4124:
4106:
4099:
4074:
4070:
4064:
4060:
4056:
4053:
4050:
4047:
4044:
4038:
4035:
4032:
4029:
4026:
4019:
4010:
4006:
4000:
3997:
3994:
3990:
3986:
3983:
3980:
3977:
3973:
3969:
3965:
3960:
3956:
3950:
3947:
3944:
3940:
3917:
3914:
3911:
3907:
3901:
3897:
3893:
3867:
3864:
3861:
3857:
3851:
3847:
3843:
3824:indexed family
3814:Direct product
3812:Main article:
3809:
3806:
3795:empty function
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3675:
3672:
3667:
3663:
3655:
3648:
3643:
3639:
3635:
3632:
3629:
3624:
3620:
3616:
3613:
3610:
3605:
3599:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3570:
3565:
3561:
3538:
3534:
3476:is the set of
3428:
3422:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3386:
3383:
3380:
3364:
3360:
3356:
3353:
3350:
3347:
3344:
3337:
3328:
3324:
3320:
3317:
3314:
3309:
3305:
3301:
3298:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3256:
3249:
3224:
3215:
3208:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3135:
3131:
3127:
3122:
3118:
3114:
3111:
3106:
3102:
3098:
3095:
3092:
3087:
3083:
3079:
3076:
3073:
3068:
3064:
3060:
3057:
3054:
3049:
3045:
3029:
3022:
3003:
2997:
2995:
2992:
2963:
2962:
2907:In this case,
2905:
2904:
2864:
2863:
2803:
2800:
2788:
2785:
2782:
2779:
2771:
2768:
2765:
2760:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2729:
2726:
2723:
2720:
2717:
2702:
2701:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2661:
2658:
2655:
2610:
2606:
2585:
2580:
2574:
2570:
2566:
2563:
2559:
2555:
2551:
2547:
2544:
2539:
2535:
2530:
2526:
2522:
2516:
2512:
2508:
2503:
2499:
2494:
2490:
2485:
2481:
2477:
2474:
2471:
2468:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2404:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2336:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2268:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2244:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1735:
1728:
1727:
1639:
1632:
1631:
1486:
1479:
1478:
1477:
1476:
1475:
1467:
1464:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1356:
1355:
1354:
1353:
1341:
1316:
1315:
1314:
1313:
1280:
1279:
1278:
1277:
1265:
1234:
1233:
1219:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1126:
1123:
1076:
1054:
1051:
1048:
1028:
1025:
1022:
1019:
1016:
1013:
1008:
1003:
998:
976:
973:
970:
967:
964:
944:
941:
938:
935:
932:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
869:Main article:
866:
863:
849:
827:
823:
819:
791:René Descartes
770:
767:
679:
676:
674:
671:
659:
656:
653:
650:
647:
644:
641:
638:
635:
632:
629:
626:
623:
620:
614:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
579:
574:
569:
564:
561:
558:
555:
552:
549:
546:
526:
506:
480:
458:
455:
452:
449:
446:
443:
438:
433:
428:
406:
403:
400:
397:
394:
391:
388:
385:
382:
379:
376:
356:
353:
350:
347:
344:
320:
300:
284:
281:
277:direct product
269:René Descartes
262:indexed family
203:
200:
197:
194:
191:
175:
172:
169:
166:
163:
160:
157:
154:
151:
148:
145:
142:
139:
136:
123:. In terms of
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8127:
8116:
8113:
8111:
8108:
8107:
8105:
8092:
8091:
8086:
8078:
8072:
8069:
8067:
8064:
8062:
8059:
8057:
8054:
8050:
8047:
8046:
8045:
8042:
8040:
8037:
8035:
8032:
8030:
8026:
8023:
8021:
8018:
8016:
8013:
8011:
8008:
8006:
8003:
8002:
8000:
7996:
7990:
7987:
7985:
7982:
7980:
7979:Recursive set
7977:
7975:
7972:
7970:
7967:
7965:
7962:
7960:
7957:
7953:
7950:
7948:
7945:
7943:
7940:
7938:
7935:
7933:
7930:
7929:
7928:
7925:
7923:
7920:
7918:
7915:
7913:
7910:
7908:
7905:
7903:
7900:
7899:
7897:
7895:
7891:
7885:
7882:
7880:
7877:
7875:
7872:
7870:
7867:
7865:
7862:
7860:
7857:
7855:
7852:
7848:
7845:
7843:
7840:
7838:
7835:
7834:
7833:
7830:
7828:
7825:
7823:
7820:
7818:
7815:
7813:
7810:
7808:
7805:
7801:
7798:
7797:
7796:
7793:
7789:
7788:of arithmetic
7786:
7785:
7784:
7781:
7777:
7774:
7772:
7769:
7767:
7764:
7762:
7759:
7757:
7754:
7753:
7752:
7749:
7745:
7742:
7740:
7737:
7736:
7735:
7732:
7731:
7729:
7727:
7723:
7717:
7714:
7712:
7709:
7707:
7704:
7702:
7699:
7696:
7695:from ZFC
7692:
7689:
7687:
7684:
7678:
7675:
7674:
7673:
7670:
7668:
7665:
7663:
7660:
7659:
7658:
7655:
7653:
7650:
7648:
7645:
7643:
7640:
7638:
7635:
7633:
7630:
7628:
7625:
7624:
7622:
7620:
7616:
7606:
7605:
7601:
7600:
7595:
7594:non-Euclidean
7592:
7588:
7585:
7583:
7580:
7578:
7577:
7573:
7572:
7570:
7567:
7566:
7564:
7560:
7556:
7553:
7551:
7548:
7547:
7546:
7542:
7538:
7535:
7534:
7533:
7529:
7525:
7522:
7520:
7517:
7515:
7512:
7510:
7507:
7505:
7502:
7500:
7497:
7496:
7494:
7490:
7489:
7487:
7482:
7476:
7471:Example
7468:
7460:
7455:
7454:
7453:
7450:
7448:
7445:
7441:
7438:
7436:
7433:
7431:
7428:
7426:
7423:
7422:
7421:
7418:
7416:
7413:
7411:
7408:
7406:
7403:
7399:
7396:
7394:
7391:
7390:
7389:
7386:
7382:
7379:
7377:
7374:
7372:
7369:
7367:
7364:
7363:
7362:
7359:
7357:
7354:
7350:
7347:
7345:
7342:
7340:
7337:
7336:
7335:
7332:
7328:
7325:
7323:
7320:
7318:
7315:
7313:
7310:
7308:
7305:
7303:
7300:
7299:
7298:
7295:
7293:
7290:
7288:
7285:
7283:
7280:
7276:
7273:
7271:
7268:
7266:
7263:
7261:
7258:
7257:
7256:
7253:
7251:
7248:
7246:
7243:
7241:
7238:
7234:
7231:
7229:
7228:by definition
7226:
7225:
7224:
7221:
7217:
7214:
7213:
7212:
7209:
7207:
7204:
7202:
7199:
7197:
7194:
7192:
7189:
7188:
7185:
7182:
7180:
7176:
7171:
7165:
7161:
7151:
7148:
7146:
7143:
7141:
7138:
7136:
7133:
7131:
7128:
7126:
7123:
7121:
7118:
7116:
7115:Kripke–Platek
7113:
7111:
7108:
7104:
7101:
7099:
7096:
7095:
7094:
7091:
7090:
7088:
7084:
7076:
7073:
7072:
7071:
7068:
7066:
7063:
7059:
7056:
7055:
7054:
7051:
7049:
7046:
7044:
7041:
7039:
7036:
7034:
7031:
7028:
7024:
7020:
7017:
7013:
7010:
7008:
7005:
7003:
7000:
6999:
6998:
6994:
6991:
6990:
6988:
6986:
6982:
6978:
6970:
6967:
6965:
6962:
6960:
6959:constructible
6957:
6956:
6955:
6952:
6950:
6947:
6945:
6942:
6940:
6937:
6935:
6932:
6930:
6927:
6925:
6922:
6920:
6917:
6915:
6912:
6910:
6907:
6905:
6902:
6900:
6897:
6895:
6892:
6891:
6889:
6887:
6882:
6874:
6871:
6869:
6866:
6864:
6861:
6859:
6856:
6854:
6851:
6849:
6846:
6845:
6843:
6839:
6836:
6834:
6831:
6830:
6829:
6826:
6824:
6821:
6819:
6816:
6814:
6811:
6809:
6805:
6801:
6799:
6796:
6792:
6789:
6788:
6787:
6784:
6783:
6780:
6777:
6775:
6771:
6761:
6758:
6756:
6753:
6751:
6748:
6746:
6743:
6741:
6738:
6736:
6733:
6729:
6726:
6725:
6724:
6721:
6717:
6712:
6711:
6710:
6707:
6706:
6704:
6702:
6698:
6690:
6687:
6685:
6682:
6680:
6677:
6676:
6675:
6672:
6670:
6667:
6665:
6662:
6660:
6657:
6655:
6652:
6650:
6647:
6645:
6642:
6641:
6639:
6637:
6636:Propositional
6633:
6627:
6624:
6622:
6619:
6617:
6614:
6612:
6609:
6607:
6604:
6602:
6599:
6595:
6592:
6591:
6590:
6587:
6585:
6582:
6580:
6577:
6575:
6572:
6570:
6567:
6565:
6564:Logical truth
6562:
6560:
6557:
6556:
6554:
6552:
6548:
6545:
6543:
6539:
6533:
6530:
6528:
6525:
6523:
6520:
6518:
6515:
6513:
6510:
6508:
6504:
6500:
6496:
6494:
6491:
6489:
6486:
6484:
6480:
6477:
6476:
6474:
6472:
6466:
6461:
6455:
6452:
6450:
6447:
6445:
6442:
6440:
6437:
6435:
6432:
6430:
6427:
6425:
6422:
6420:
6417:
6415:
6412:
6410:
6407:
6405:
6402:
6400:
6397:
6393:
6390:
6389:
6388:
6385:
6384:
6382:
6378:
6374:
6367:
6362:
6360:
6355:
6353:
6348:
6347:
6344:
6332:
6331:Ernst Zermelo
6329:
6327:
6324:
6322:
6319:
6317:
6316:Willard Quine
6314:
6312:
6309:
6307:
6304:
6302:
6299:
6297:
6294:
6292:
6289:
6287:
6284:
6282:
6279:
6277:
6274:
6273:
6271:
6269:
6268:Set theorists
6265:
6259:
6256:
6254:
6251:
6249:
6246:
6245:
6243:
6237:
6235:
6232:
6231:
6228:
6220:
6217:
6215:
6214:Kripke–Platek
6212:
6208:
6205:
6204:
6203:
6200:
6199:
6198:
6195:
6191:
6188:
6187:
6186:
6185:
6181:
6177:
6174:
6173:
6172:
6169:
6168:
6165:
6162:
6160:
6157:
6155:
6152:
6150:
6147:
6146:
6144:
6140:
6134:
6131:
6129:
6126:
6124:
6121:
6119:
6117:
6112:
6110:
6107:
6105:
6102:
6099:
6095:
6092:
6090:
6087:
6083:
6080:
6078:
6075:
6073:
6070:
6069:
6068:
6065:
6062:
6058:
6055:
6053:
6050:
6048:
6045:
6043:
6040:
6039:
6037:
6034:
6030:
6024:
6021:
6019:
6016:
6014:
6011:
6009:
6006:
6004:
6001:
5999:
5996:
5994:
5991:
5987:
5984:
5982:
5979:
5978:
5977:
5974:
5972:
5969:
5967:
5964:
5962:
5959:
5957:
5954:
5951:
5947:
5944:
5942:
5939:
5937:
5934:
5933:
5931:
5925:
5922:
5921:
5918:
5912:
5909:
5907:
5904:
5902:
5899:
5897:
5894:
5892:
5889:
5887:
5884:
5882:
5879:
5876:
5873:
5871:
5868:
5867:
5865:
5863:
5859:
5851:
5850:specification
5848:
5846:
5843:
5842:
5841:
5838:
5837:
5834:
5831:
5829:
5826:
5824:
5821:
5819:
5816:
5814:
5811:
5809:
5806:
5804:
5801:
5799:
5796:
5794:
5791:
5789:
5786:
5782:
5779:
5777:
5774:
5772:
5769:
5768:
5767:
5764:
5762:
5759:
5758:
5756:
5754:
5750:
5745:
5735:
5732:
5731:
5729:
5725:
5721:
5714:
5709:
5707:
5702:
5700:
5695:
5694:
5691:
5685:
5682:
5678:
5674:
5673:
5668:
5664:
5662:
5659:
5658:
5654:
5646:
5640:
5637:
5633:
5630:F. R. Drake,
5627:
5624:
5621:
5617:
5611:
5608:
5605:
5601:
5595:
5592:
5589:
5585:
5579:
5577:
5573:
5568:
5566:9780486652528
5562:
5558:
5557:
5549:
5546:
5531:
5524:
5521:
5508:
5504:
5498:
5495:
5482:
5478:
5474:
5468:
5466:
5464:
5460:
5448:
5444:
5437:
5434:
5429:
5425:
5418:
5415:
5402:
5401:
5396:
5393:
5386:
5384:
5382:
5378:
5371:
5367:
5364:
5362:
5359:
5357:
5354:
5352:
5351:Outer product
5349:
5347:
5344:
5342:
5339:
5337:
5334:
5332:
5331:Empty product
5329:
5327:
5324:
5321:
5318:
5317:
5313:
5311:
5309:
5305:
5287:
5283:
5279:
5261:
5257:
5251:
5247:
5240:
5236:
5228:
5224:
5216:
5212:
5208:
5204:
5199:
5194:
5190:
5178:
5174:
5166:
5164:
5162:
5158:
5154:
5153:right adjoint
5150:
5146:
5144:
5143:fiber product
5140:
5136:
5132:
5124:
5119:
5117:
5098:
5095:
5075:
5033:
5025:
5009:
5000:
4986:
4966:
4946:
4943:
4940:
4920:
4900:
4892:
4876:
4873:
4870:
4863:be a set and
4850:
4838:
4836:
4834:
4829:
4816:
4807:
4801:
4798:
4792:
4786:
4780:
4774:
4771:
4768:
4759:
4756:
4753:
4741:
4737:
4731:
4727:
4721:
4717:
4684:
4682:
4678:
4674:
4661:
4654:
4647:
4638:
4633:
4631:
4584:
4569:
4553:
4550:
4542:
4534:
4519:
4516:
4513:
4509:
4500:
4489:
4463:
4459:
4435:
4430:
4427:
4424:
4420:
4416:
4411:
4407:
4401:
4398:
4395:
4391:
4378:
4371:
4367:
4365:
4364:
4340:
4334:
4331:
4325:
4317:
4313:
4292:
4287:
4283:
4274:
4270:
4264:
4261:
4258:
4254:
4250:
4245:
4241:
4223:
4207:
4203:
4197:
4194:
4191:
4142:
4138:
4132:
4129:
4126:
4122:
4113:
4109:
4102:
4088:
4072:
4068:
4062:
4058:
4054:
4048:
4042:
4036:
4033:
4030:
4027:
4017:
4008:
4004:
3998:
3995:
3992:
3988:
3981:
3978:
3975:
3967:
3963:
3958:
3954:
3948:
3945:
3942:
3938:
3915:
3912:
3909:
3899:
3895:
3865:
3862:
3859:
3849:
3845:
3833:
3825:
3821:
3815:
3807:
3805:
3800:
3796:
3792:
3791:singleton set
3776:
3763:
3760:
3754:
3748:
3744:
3740:
3736:
3730:
3717:
3708:
3705:
3702:
3699:
3696:
3690:
3687:
3673:
3670:
3665:
3661:
3641:
3637:
3633:
3630:
3627:
3622:
3618:
3608:
3603:
3597:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3568:
3563:
3559:
3536:
3532:
3519:
3511:
3509:
3495:
3491:
3484:
3479:
3474:
3468:
3464:
3460:
3456:
3451:
3447:
3443:
3434:
3426:
3423:
3421:
3408:
3399:
3396:
3393:
3390:
3387:
3381:
3378:
3362:
3358:
3354:
3348:
3342:
3326:
3322:
3318:
3315:
3312:
3307:
3303:
3293:
3290:
3287:
3284:
3281:
3275:
3272:
3259:
3255:
3248:
3234:
3227:
3218:
3214:
3207:
3201:
3197:
3189:
3170:
3167:
3164:
3161:
3158:
3152:
3149:
3133:
3129:
3125:
3120:
3116:
3112:
3104:
3100:
3096:
3093:
3090:
3085:
3081:
3071:
3066:
3062:
3058:
3055:
3052:
3047:
3043:
3032:
3028:
3021:
3012:
3001:
2998:
2993:
2991:
2981:
2976:
2972:
2966:
2958:
2952:
2946:
2940:
2936:
2932:
2926:
2925:
2924:
2921:
2917:
2913:
2899:
2893:
2887:
2883:
2877:
2876:
2875:
2859:
2855:
2851:
2850:
2849:
2846:
2842:
2827:
2820:
2815:
2809:
2801:
2799:
2786:
2783:
2780:
2777:
2769:
2766:
2763:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2724:
2721:
2718:
2715:
2712:if both
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2659:
2656:
2653:
2641:
2640:
2639:
2637:
2632:
2626:
2608:
2604:
2583:
2578:
2572:
2568:
2564:
2561:
2557:
2553:
2549:
2545:
2542:
2537:
2533:
2528:
2524:
2520:
2514:
2510:
2506:
2501:
2497:
2492:
2488:
2483:
2475:
2472:
2469:
2443:
2437:
2434:
2431:
2419:
2416:
2413:
2407:
2405:
2397:
2391:
2385:
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2375:
2369:
2366:
2363:
2357:
2351:
2348:
2345:
2339:
2337:
2329:
2326:
2323:
2317:
2314:
2307:
2301:
2298:
2295:
2289:
2283:
2280:
2277:
2271:
2269:
2261:
2258:
2255:
2249:
2246:
2233:
2217:
2214:
2208:
2202:
2193:
2184:
2178:
2172:
2169:
2163:
2157:
2154:
2151:
2139:
2136:
2133:
2121:
2105:
2102:
2096:
2090:
2081:
2072:
2069:
2066:
2060:
2054:
2051:
2048:
2039:
2033:
2030:
2024:
2018:
2009:
2003:
2000:
1997:
1991:
1985:
1982:
1979:
1967:
1951:
1948:
1945:
1939:
1933:
1930:
1927:
1921:
1915:
1912:
1909:
1903:
1897:
1894:
1891:
1880:
1875:
1859:
1856:
1853:
1847:
1841:
1838:
1835:
1829:
1823:
1820:
1817:
1811:
1805:
1802:
1799:
1788:
1787:intersections
1774:
1770:
1764:
1760:
1753:
1749:
1745:
1741:
1732:
1720:
1716:
1710:
1706:
1699:
1695:
1691:
1687:
1681:
1675:
1665:
1656:
1647:
1636:
1625:
1621:
1615:
1611:
1606:
1602:
1598:
1594:
1588:
1584:
1578:
1574:
1569:
1565:
1561:
1552:
1548:
1542:
1538:
1531:
1527:
1523:
1513:
1503:
1494:
1483:
1473:
1465:
1463:
1459:
1455:
1451:
1444:
1440:
1436:
1428:
1408:
1405:
1402:
1396:
1393:
1390:
1387:
1384:
1378:
1375:
1372:
1361:
1350:
1346:
1342:
1338:
1334:
1330:
1329:
1326:
1322:
1318:
1317:
1310:
1306:
1302:
1298:
1294:
1293:
1290:
1286:
1282:
1281:
1274:
1270:
1266:
1262:
1258:
1254:
1253:
1250:
1243:
1239:
1238:
1237:
1236:For example:
1231:
1220:
1210:
1209:
1208:
1206:
1205:ordered pairs
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1161:
1156:
1152:
1146:
1124:
1122:
1120:
1116:
1112:
1111:specification
1108:
1104:
1100:
1096:
1092:
1052:
1049:
1046:
1020:
1017:
1014:
971:
968:
965:
936:
933:
930:
924:
918:
909:
903:
900:
897:
886:
882:
878:
872:
864:
862:
821:
800:
797:, called its
796:
792:
788:
784:
775:
768:
766:
764:
760:
755:
752:
748:
743:
724:
720:
715:
713:
712:ordered pairs
697:
693:
684:
677:
672:
670:
657:
648:
645:
642:
636:
633:
630:
627:
624:
621:
612:
609:
606:
600:
591:
588:
585:
562:
559:
553:
550:
547:
544:
524:
504:
496:
450:
447:
444:
398:
395:
392:
386:
380:
351:
348:
345:
334:
318:
298:
290:
282:
280:
278:
274:
270:
265:
263:
259:
255:
243:
231:
224:
220:
214:
201:
195:
192:
189:
173:
170:
167:
164:
158:
155:
152:
143:
140:
137:
134:
126:
104:
100:
95:
94:ordered pairs
90:
86:
74:
70:
66:
62:
55:} and {1,2,3}
54:
50:
46:
41:
35:
30:
19:
8081:
7879:Ultraproduct
7726:Model theory
7691:Independence
7627:Formal proof
7619:Proof theory
7602:
7575:
7532:real numbers
7504:second-order
7415:Substitution
7292:Metalanguage
7233:conservative
7206:Axiom schema
7150:Constructive
7120:Morse–Kelley
7086:Set theories
7065:Aleph number
7058:inaccessible
6964:Grothendieck
6862:
6848:intersection
6735:Higher-order
6723:Second-order
6669:Truth tables
6626:Venn diagram
6409:Formal proof
6281:Georg Cantor
6276:Paul Bernays
6207:Morse–Kelley
6182:
6115:
6114:Subset
6061:hereditarily
6023:Venn diagram
5981:ordered pair
5896:Intersection
5869:
5840:Axiom schema
5670:
5647:. MIT Press.
5644:
5639:
5631:
5626:
5615:
5610:
5599:
5594:
5583:
5555:
5548:
5536:. Retrieved
5523:
5511:. Retrieved
5506:
5497:
5487:September 5,
5485:. Retrieved
5481:the original
5476:
5452:September 5,
5450:. Retrieved
5447:Math Insight
5446:
5436:
5430:. p. 6.
5423:
5417:
5406:September 5,
5404:. Retrieved
5398:
5366:Product type
5285:
5281:
5277:
5259:
5255:
5249:
5245:
5238:
5234:
5226:
5222:
5214:
5210:
5206:
5202:
5192:
5188:
5173:graph theory
5170:
5167:Graph theory
5157:final object
5147:
5128:
5001:
4890:
4842:
4830:
4739:
4735:
4729:
4725:
4719:
4715:
4688:
4676:
4672:
4659:
4652:
4645:
4642:
4495:
4457:
4373:
4369:
4368:
4357:
4224:
4104:
4097:
3826:of sets. If
3817:
3764:
3758:
3752:
3746:
3742:
3738:
3734:
3731:
3514:
3512:
3493:
3489:
3482:
3478:real numbers
3472:
3466:
3462:
3458:
3449:
3445:
3441:
3432:
3430:
3424:
3257:
3253:
3246:
3232:
3231:{1, 2, ...,
3222:
3216:
3212:
3205:
3190:
3030:
3026:
3019:
3007:
3005:
2999:
2974:
2970:
2967:
2964:
2956:
2950:
2944:
2938:
2934:
2930:
2922:
2915:
2911:
2906:
2897:
2891:
2885:
2881:
2865:
2857:
2853:
2844:
2840:
2825:
2818:
2811:
2732:, then
2703:
2664:, then
2633:
2623:denotes the
2234:
2122:
1968:
1876:
1784:
1772:
1768:
1762:
1758:
1751:
1747:
1743:
1739:
1718:
1714:
1708:
1704:
1697:
1693:
1689:
1685:
1673:
1663:
1654:
1645:
1642:
1640:Example sets
1623:
1619:
1613:
1609:
1604:
1600:
1596:
1586:
1582:
1576:
1572:
1567:
1563:
1559:
1550:
1546:
1540:
1536:
1529:
1525:
1521:
1511:
1501:
1492:
1489:
1487:Example sets
1457:
1453:
1449:
1442:
1438:
1434:
1426:
1357:
1348:
1344:
1336:
1332:
1324:
1320:
1308:
1304:
1300:
1296:
1288:
1284:
1272:
1268:
1260:
1256:
1248:
1241:
1235:
1214:is equal to
1203:because the
1154:
1150:
1147:
1128:
881:ordered pair
874:
795:real numbers
780:
756:
750:
746:
744:
738:), (2,
730:), (A,
722:
718:
716:
689:
493:denotes the
286:
266:
238:
232:
222:
218:
215:
102:
98:
88:
84:
68:
58:
52:
48:
44:
29:
7989:Type theory
7937:undecidable
7869:Truth value
7756:equivalence
7435:non-logical
7048:Enumeration
7038:Isomorphism
6985:cardinality
6969:Von Neumann
6934:Ultrafilter
6899:Uncountable
6833:equivalence
6750:Quantifiers
6740:Fixed-point
6709:First-order
6589:Consistency
6574:Proposition
6551:Traditional
6522:Lindström's
6512:Compactness
6454:Type theory
6399:Cardinality
6306:Thomas Jech
6149:Alternative
6128:Uncountable
6082:Ultrafilter
5941:Cardinality
5845:replacement
5793:Determinacy
5513:December 1,
5503:"Cartesian"
4889:. Then the
4634:Other forms
4305:defined by
3035:as the set
2965:and so on.
2923:Similarly,
2830:. Both set
2814:cardinality
2802:Cardinality
1360:associative
1160:commutative
799:coordinates
61:mathematics
8104:Categories
7800:elementary
7493:arithmetic
7361:Quantifier
7339:functional
7211:Expression
6929:Transitive
6873:identities
6858:complement
6791:hereditary
6774:Set theory
6301:Kurt Gödel
6286:Paul Cohen
6123:Transitive
5891:Identities
5875:Complement
5862:Operations
5823:Regularity
5761:Adjunction
5720:Set theory
5528:Corry, S.
5372:References
5002:Normally,
3775:isomorphic
3524:, denoted
3240:to be the
2982:if either
2919:| = 4
2806:See also:
1470:See also:
335:of a pair
127:, that is
82:, denoted
65:set theory
8071:Supertask
7974:Recursion
7932:decidable
7766:saturated
7744:of models
7667:deductive
7662:axiomatic
7582:Hilbert's
7569:Euclidean
7550:canonical
7473:axiomatic
7405:Signature
7334:Predicate
7223:Extension
7145:Ackermann
7070:Operation
6949:Universal
6939:Recursive
6914:Singleton
6909:Inhabited
6894:Countable
6884:Types of
6868:power set
6838:partition
6755:Predicate
6701:Predicate
6616:Syllogism
6606:Soundness
6579:Inference
6569:Tautology
6471:paradoxes
6234:Paradoxes
6154:Axiomatic
6133:Universal
6109:Singleton
6104:Recursive
6047:Countable
6042:Amorphous
5901:Power set
5818:Power set
5776:dependent
5771:countable
5677:EMS Press
5600:ProofWiki
5400:MathWorld
5099:×
4944:×
4874:⊆
4757:×
4585:ω
4554:⋯
4551:×
4543:×
4525:∞
4510:∏
4428:∈
4421:∏
4399:∈
4392:∏
4314:π
4280:→
4262:∈
4255:∏
4242:π
4225:For each
4195:∈
4130:∈
4123:∏
4087:index set
4055:∈
4031:∈
4025:∀
3996:∈
3989:⋃
3985:→
3946:∈
3939:∏
3913:∈
3863:∈
3832:index set
3703:…
3691:∈
3681:for every
3671:∈
3631:…
3598:⏟
3591:×
3588:⋯
3585:×
3579:×
3520:of a set
3435:of a set
3394:…
3382:∈
3372:for every
3355:∈
3319:∪
3316:⋯
3313:∪
3300:→
3288:…
3165:…
3153:∈
3143:for every
3126:∈
3113:∣
3094:…
3059:×
3056:⋯
3053:×
2781:⊆
2767:⊆
2759:⟺
2751:×
2745:⊆
2739:×
2728:∅
2725:≠
2683:×
2677:⊆
2671:×
2657:⊆
2609:∁
2573:∁
2565:×
2554:∪
2543:×
2538:∁
2525:∪
2515:∁
2507:×
2502:∁
2484:∁
2473:×
2435:×
2426:∖
2417:×
2395:∖
2386:×
2367:×
2358:∪
2349:×
2327:∪
2318:×
2299:×
2290:∩
2281:×
2259:∩
2250:×
2215:×
2206:∖
2194:∪
2182:∖
2173:×
2155:×
2146:∖
2137:×
2103:×
2094:∖
2082:∪
2070:∪
2061:×
2052:∩
2040:∪
2031:×
2022:∖
2001:×
1992:∪
1983:×
1949:×
1940:∪
1931:×
1922:≠
1913:∪
1904:×
1895:∪
1857:×
1848:∩
1839:×
1821:∩
1812:×
1803:∩
1406:×
1397:×
1391:≠
1385:×
1376:×
1323:= {1,2};
1230:empty set
1185:×
1179:≠
1173:×
1145:be sets.
1119:relations
1115:functions
1107:power set
1091:power set
1050:×
1018:∪
822:×
763:bijection
625:∈
619:∃
610:∈
604:∃
601:∣
589:∪
563:∈
548:×
495:power set
448:∪
264:of sets.
193:∈
171:∈
165:∣
138:×
8056:Logicism
8049:timeline
8025:Concrete
7884:Validity
7854:T-schema
7847:Kripke's
7842:Tarski's
7837:semantic
7827:Strength
7776:submodel
7771:spectrum
7739:function
7587:Tarski's
7576:Elements
7563:geometry
7519:Robinson
7440:variable
7425:function
7398:spectrum
7388:Sentence
7344:variable
7287:Language
7240:Relation
7201:Automata
7191:Alphabet
7175:language
7029:-jection
7007:codomain
6993:Function
6954:Universe
6924:Infinite
6828:Relation
6611:Validity
6601:Argument
6499:theorem,
6238:Problems
6142:Theories
6118:Superset
6094:Infinite
5923:Concepts
5803:Infinity
5727:Overview
5314:See also
5196:, whose
5024:universe
4891:cylinder
4839:Cylinder
3820:infinite
3799:codomain
3252:× ... ×
3211:× ... ×
2980:infinite
2968:The set
2834:and set
2828:= {5, 6}
2821:= {a, b}
2650:if
1113:. Since
759:disjoint
673:Examples
7998:Related
7795:Diagram
7693: (
7672:Hilbert
7657:Systems
7652:Theorem
7530:of the
7475:systems
7255:Formula
7250:Grammar
7166: (
7110:General
6823:Forcing
6808:Element
6728:Monadic
6503:paradox
6444:Theorem
6380:General
6176:General
6171:Zermelo
6077:subbase
6059: (
5998:Forcing
5976:Element
5948: (
5926:Methods
5813:Pairing
5679:, 2001
5304:product
5159:) is a
5151:is the
5135:product
3830:is any
3750:, with
3196:-tuples
3025:, ...,
2636:subsets
1431:, then
1291:= {1,2}
1251:= {3,4}
1244:= {1,2}
1228:is the
1158:is not
1099:pairing
839:, with
223:columns
71:of two
7761:finite
7524:Skolem
7477:
7452:Theory
7420:Symbol
7410:String
7393:atomic
7270:ground
7265:closed
7260:atomic
7216:ground
7179:syntax
7075:binary
7002:domain
6919:Finite
6684:finite
6542:Logics
6501:
6449:Theory
6067:Filter
6057:Finite
5993:Family
5936:Almost
5781:global
5766:Choice
5753:Axioms
5563:
5538:May 5,
5509:. 2009
5198:vertex
5175:, the
4833:tuples
4568:vector
4486:, the
4040:
4022:
4014:
3834:, and
3685:
3677:
3658:
3650:
3498:where
3470:where
3376:
3368:
3340:
3332:
3147:
3139:
2960:|
2928:|
2909:|
2901:|
2879:|
2596:where
1141:, and
1109:, and
1039:, and
694:. The
616:
469:where
187:
177:
119:is in
111:is in
107:where
67:, the
7751:Model
7499:Peano
7356:Proof
7196:Arity
7125:Naive
7012:image
6944:Fuzzy
6904:Empty
6853:union
6798:Class
6439:Model
6429:Lemma
6387:Axiom
6159:Naive
6089:Fuzzy
6052:Empty
6035:types
5986:tuple
5956:Class
5950:large
5911:Union
5828:Union
5533:(PDF)
5298:′ in
5272:′ in
4743:with
4665:, ...
4599:, or
3797:with
3455:plane
3017:sets
3013:over
2638:are:
1879:union
1766:) ∪ (
1746:) × (
1712:) ∩ (
1692:) × (
1617:) \ (
1607:) = (
1592:, and
1580:) ∪ (
1570:) = (
1544:) ∩ (
1507:, and
1429:= {1}
1103:union
887:, is
751:Ranks
747:Suits
723:Suits
719:Ranks
254:tuple
7874:Type
7677:list
7481:list
7458:list
7447:Term
7381:rank
7275:open
7169:list
6981:Maps
6886:sets
6745:Free
6715:list
6465:list
6392:list
6072:base
5561:ISBN
5540:2023
5515:2009
5489:2020
5454:2020
5408:2020
5290:and
5264:and
5231:and
5209:) ×
5182:and
4979:and
4843:Let
4701:and
4361:-th
3765:The
3513:The
3502:and
3431:The
3221:) ×
2823:and
2812:The
1441:) ×
1218:, or
1129:Let
805:and
700:{♠,
517:and
311:and
219:rows
115:and
78:and
73:sets
7561:of
7543:of
7491:of
7023:Sur
6997:Map
6804:Ur-
6786:Set
6033:Set
5171:In
5088:is
4959:of
4893:of
4733:to
4709:to
4697:to
4689:If
4452:to
4229:in
3773:is
3510:).
3191:of
2986:or
2978:is
2627:of
1756:≠ (
1702:= (
1599:× (
1562:× (
1534:= (
1524:× (
1452:× (
1327:= ∅
1224:or
1095:ZFC
785:in
708:, ♣
367:as
59:In
8106::
7947:NP
7571::
7565::
7495::
7172:),
7027:Bi
7019:In
5675:,
5669:,
5575:^
5505:.
5475:.
5462:^
5445:.
5426:.
5397:.
5380:^
5310:.
5284:=
5278:or
5276:,
5258:=
5248:×
5241:′)
5237:′,
5191:×
5163:.
5145:.
5116:.
4999:.
4738:×
4728:×
4718:×
4681:.
4658:,
4651:,
4630:.
4366:.
4222:.
3822:)
3804:.
3762:.
3745:×
3741:×
3737:=
3480::
3465:×
3461:=
3448:×
3444:=
3219:−1
2973:×
2937:×
2933:×
2914:×
2884:×
2856:×
2843:×
2631:.
1677:=
1667:=
1660:,
1658:=
1651:,
1649:=
1515:=
1505:=
1498:,
1496:=
1462:.
1456:×
1437:×
1347:×
1335:×
1307:×
1303:=
1299:×
1287:=
1271:×
1259:×
1246:;
1162:,
1153:×
1137:,
1133:,
1105:,
1101:,
749:×
721:×
704:,
279:.
230:.
221:×
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