45:
4499:
2752:
4433:
1503:
The use of representatives for representing classes allows avoiding to consider explicitly classes as sets. In this case, the canonical surjection that maps an element to its class is replaced by the function that maps an element to the representative of its class. In the preceding example, this
51:
is an example of an equivalence relation. The leftmost two triangles are congruent, while the third and fourth triangles are not congruent to any other triangle shown here. Thus, the first two triangles are in the same equivalence class, while the third and fourth triangles are each in their own
2247:. This relation gives rise to exactly two equivalence classes: one class consists of all even numbers, and the other class consists of all odd numbers. Using square brackets around one member of the class to denote an equivalence class under this relation,
3983:
Although the term can be used for any equivalence relation's set of equivalence classes, possibly with further structure, the intent of using the term is generally to compare that type of equivalence relation on a set
3640:
3281:
3923:
formed on the set of equivalence classes of an equivalence relation on a topological space, using the original space's topology to create the topology on the set of equivalence classes.
3980:
A normal subgroup of a topological group, acting on the group by translation action, is a quotient space in the senses of topology, abstract algebra, and group actions simultaneously.
2347:
2057:
978:
2623:
and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers. The same construction can be generalized to the
1475:
3324:
3575:
2520:
3977:
of a subgroup of a group, which arise from the action of the subgroup on the group by left translations, or respectively the left cosets as orbits under right translation.
2189:
1535:
3535:
3372:
3129:
888:
463:
430:
3877:
3830:
3469:
3447:
3425:
2922:
2450:
2108:
1855:
1200:
600:
515:
201:
4956:
3507:
3399:
3157:
3023:
2880:
765:
327:
2682:
2215:
1970:
1790:
850:
824:
798:
733:
707:
649:
2555:
675:
350:
3774:
3712:
2587:
2405:
3080:
3051:
2974:
2621:
1430:
1345:
1251:
1075:
2241:
1306:
4028:
4005:
3801:
2945:
2811:
2788:
2428:
2286:
2154:
2131:
1941:
1878:
1498:
377:
224:
3897:
3850:
3680:
3660:
3344:
3219:
3199:
3179:
3100:
2994:
2851:
2831:
2722:
2702:
2652:
2470:
2369:
2086:
2009:
1918:
1898:
1830:
1810:
1764:
1744:
1724:
1700:
1624:
1595:
1575:
1220:
1165:
1145:
1117:
1095:
1044:
1024:
916:
620:
575:
485:
405:
268:
248:
179:
149:
129:
105:
81:
3738:
2312:
1676:
1650:
294:
3966:, or any quotient algebra. However, the use of the term for the more general cases can as often be by analogy with the orbits of a group action.
4259:
4223:
379:
meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the
4645:
4465:
4072:
4973:
5116:
2887:
4007:
either to an equivalence relation that induces some structure on the set of equivalence classes from a structure of the same kind on
4341:
4305:
4241:
4205:
3973:
on a set may be called the quotient space of the action on the set, particularly when the orbits of the group action are the right
4951:
3934:
on the underlying set of an algebra allow the algebra to induce an algebra on the equivalence classes of the relation, called a
4054:
based on dividing the possible program inputs into equivalence classes according to the behavior of the program on those inputs
3943:
518:
3580:
3224:
517:
is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examples include
4437:
4725:
4604:
4968:
4030:
or to the orbits of a group action. Both the sense of a structure preserved by an equivalence relation, and the study of
4066:
3935:
4961:
1358:
Sometimes, there is a section that is more "natural" than the other ones. In this case, the representatives are called
4599:
4562:
4084:
3970:
5111:
4650:
4542:
4530:
4525:
3916:
1348:
522:
31:
4458:
4047:
5070:
4988:
4863:
4815:
4629:
4552:
4031:
3103:
38:
2317:
2015:
5022:
4903:
4715:
4535:
3475:
924:
4938:
4852:
4772:
4752:
4730:
2891:
48:
1435:
3286:
44:
5012:
5002:
4836:
4767:
4720:
4660:
4547:
3777:
3540:
2475:
1313:
1097:
554:
84:
5121:
5007:
4918:
4831:
4826:
4821:
4635:
4577:
4515:
4451:
4386:
3931:
2729:
2169:
1507:
1381:
996:
891:
488:
4930:
4925:
4710:
4665:
4572:
2764:
2624:
2159:
1703:
1538:
1366:
1277:
768:
678:
353:
3515:
3352:
3109:
2728:, then the set of lines that are parallel to each other form an equivalence class, as long as a
855:
435:
410:
3855:
3808:
3452:
3430:
3408:
2905:
2433:
2091:
1838:
1173:
583:
498:
184:
4787:
4624:
4616:
4587:
4557:
4488:
4337:
4301:
4255:
4237:
4219:
4201:
4057:
3920:
3480:
3380:
3138:
2999:
2856:
2733:
1000:
992:
984:
983:
The word "class" in the term "equivalence class" may generally be considered as a synonym of "
738:
542:
530:
492:
299:
61:
2661:
2194:
1949:
1769:
829:
803:
777:
712:
686:
628:
5106:
5075:
5065:
5050:
5045:
4913:
4567:
4293:
4051:
3927:
3402:
2760:
2525:
654:
332:
3747:
3685:
2560:
2378:
4944:
4882:
4700:
4520:
3955:
3056:
3027:
2950:
2655:
2628:
2595:
2590:
1406:
1352:
1319:
1225:
1049:
578:
538:
155:
2220:
1283:
4010:
3987:
3783:
2927:
2793:
2770:
2410:
2250:
2136:
2113:
1923:
1860:
1480:
359:
206:
5080:
4877:
4858:
4762:
4747:
4704:
4640:
4582:
3963:
3947:
3939:
3882:
3835:
3805:
More generally, a function may map equivalent arguments (under an equivalence relation
3665:
3645:
3329:
3204:
3184:
3164:
3085:
2979:
2836:
2816:
2725:
2707:
2687:
2637:
2455:
2354:
2071:
1976:
1903:
1883:
1815:
1795:
1749:
1729:
1709:
1685:
1600:
1580:
1560:
1360:
1205:
1168:
1150:
1130:
1102:
1080:
1029:
1009:
901:
605:
560:
526:
470:
390:
253:
233:
164:
134:
114:
90:
66:
3717:
2291:
1655:
1629:
273:
5100:
5085:
5055:
4887:
4801:
4796:
3959:
3741:
2883:
2746:
1679:
534:
2110:
the equivalence relation "has the same area as", then for each positive real number
352:
The definition of equivalence relations implies that the equivalence classes form a
5035:
5030:
4848:
4777:
4735:
4594:
4498:
3132:
2372:
988:
1268:
Every element of an equivalence class characterizes the class, and may be used to
2882:
Among these graphs are the graphs of equivalence relations. These graphs, called
5060:
4695:
2244:
57:
30:
This article is about equivalency in mathematics. For equivalency in music, see
5040:
4908:
4811:
4474:
3951:
1746:
belongs to one and only one equivalence class. Conversely, every partition of
3954:. By extension, in abstract algebra, the term quotient space may be used for
4843:
4806:
4757:
4655:
4061:
17:
4432:
2751:
3912:
3900:
2133:
there will be an equivalence class of all the rectangles that have area
4081: – Mathematical construction of a set with an equivalence relation
2163:
1370:
1276:
of the class. The choice of a representative in each class defines an
4868:
4690:
4078:
4216:
Sets, Functions, and Logic: An
Introduction to Abstract Mathematics
1766:
comes from an equivalence relation in this way, according to which
1387:
is an equivalence relation on the integers, for which two integers
4740:
4507:
3974:
37:"Quotient map" redirects here. For Quotient map in topology, see
1253:
which maps each element to its equivalence class, is called the
4447:
1835:
It follows from the properties in the previous section that if
1477:
Each class contains a unique non-negative integer smaller than
4075: – Generalization of equivalence classes to scheme theory
4443:
1515:
111:. These equivalence classes are constructed so that elements
4298:
Chapter Zero: Fundamental
Notions of Abstract Mathematics
4087: – Set that intersects every one of a family of sets
2732:. In this situation, each equivalence class determines a
3635:{\displaystyle f\left(x_{1}\right)=f\left(x_{2}\right).}
4116:
4114:
4112:
3276:{\displaystyle f\left(x_{1}\right)=f\left(x_{2}\right)}
1500:
and these integers are the canonical representatives.
4252:
Proof, Logic and
Conjecture: A Mathematician's Toolbox
3852:) to equivalent values (under an equivalence relation
1438:
987:", although some equivalence classes are not sets but
4013:
3990:
3885:
3858:
3838:
3811:
3786:
3750:
3720:
3688:
3668:
3648:
3583:
3543:
3518:
3483:
3455:
3433:
3411:
3383:
3355:
3332:
3289:
3227:
3207:
3187:
3167:
3141:
3112:
3088:
3059:
3030:
3002:
2982:
2953:
2930:
2908:
2859:
2839:
2819:
2796:
2773:
2710:
2690:
2664:
2640:
2598:
2563:
2528:
2478:
2458:
2436:
2413:
2381:
2357:
2320:
2294:
2253:
2223:
2197:
2172:
2139:
2116:
2094:
2074:
2018:
1979:
1952:
1926:
1906:
1886:
1863:
1841:
1818:
1798:
1772:
1752:
1732:
1712:
1688:
1658:
1632:
1603:
1583:
1563:
1510:
1483:
1409:
1322:
1286:
1228:
1208:
1176:
1153:
1133:
1105:
1083:
1052:
1032:
1012:
927:
904:
858:
832:
806:
780:
741:
715:
689:
657:
631:
608:
586:
563:
501:
473:
438:
413:
393:
362:
335:
302:
276:
256:
236:
209:
187:
167:
137:
117:
93:
69:
3405:
of finite groups. Some authors use "compatible with
5021:
4984:
4896:
4786:
4674:
4615:
4506:
4481:
1682:. Therefore, the set of all equivalence classes of
1272:it. When such an element is chosen, it is called a
4069: – Mathematical concept for comparing objects
4022:
3999:
3891:
3871:
3844:
3824:
3795:
3768:
3732:
3706:
3674:
3654:
3634:
3569:
3529:
3501:
3463:
3441:
3419:
3393:
3366:
3338:
3318:
3275:
3213:
3193:
3173:
3151:
3123:
3094:
3074:
3045:
3017:
2988:
2968:
2939:
2916:
2874:
2845:
2825:
2805:
2782:
2716:
2696:
2676:
2646:
2615:
2581:
2549:
2514:
2464:
2444:
2422:
2399:
2363:
2341:
2306:
2280:
2235:
2209:
2183:
2148:
2125:
2102:
2080:
2051:
2003:
1964:
1935:
1912:
1892:
1872:
1849:
1824:
1804:
1784:
1758:
1738:
1718:
1694:
1670:
1644:
1618:
1589:
1569:
1529:
1492:
1469:
1424:
1339:
1300:
1245:
1214:
1194:
1159:
1139:
1111:
1089:
1069:
1038:
1018:
972:
910:
882:
844:
818:
792:
759:
727:
701:
669:
643:
614:
594:
569:
509:
479:
457:
424:
399:
371:
344:
321:
288:
262:
242:
218:
195:
173:
143:
123:
99:
75:
4379:Mathematical Thinking: Problem Solving and Proofs
4316:The Structure of Proof: With Logic and Set Theory
1316:with the canonical surjection is the identity of
2886:, are characterized as the graphs such that the
4218:(3rd ed.), Chapman & Hall/ CRC Press,
4034:under group actions, lead to the definition of
3903:of sets equipped with an equivalence relation.
83:have a notion of equivalence (formalized as an
2755:Graph of an example equivalence with 7 classes
4459:
2088:be the set of all rectangles in a plane, and
8:
964:
940:
4466:
4452:
4444:
4168:
4156:
3776:This equivalence relation is known as the
4370:An Introduction to Mathematical Reasoning
4277:Mathematical Reasoning: Writing and Proof
4012:
3989:
3884:
3863:
3857:
3837:
3816:
3810:
3785:
3749:
3719:
3687:
3667:
3647:
3619:
3595:
3582:
3561:
3548:
3542:
3523:
3519:
3517:
3482:
3460:
3456:
3454:
3438:
3434:
3432:
3416:
3412:
3410:
3384:
3382:
3360:
3356:
3354:
3331:
3307:
3294:
3288:
3263:
3239:
3226:
3206:
3186:
3166:
3142:
3140:
3117:
3113:
3111:
3087:
3058:
3029:
3001:
2981:
2952:
2929:
2913:
2909:
2907:
2858:
2838:
2818:
2795:
2772:
2709:
2689:
2663:
2639:
2602:
2597:
2562:
2527:
2477:
2457:
2441:
2437:
2435:
2412:
2380:
2356:
2331:
2326:
2322:
2321:
2319:
2293:
2252:
2222:
2196:
2174:
2173:
2171:
2138:
2115:
2099:
2095:
2093:
2073:
2017:
1978:
1951:
1943:the following statements are equivalent:
1925:
1905:
1885:
1862:
1846:
1842:
1840:
1832:belong to the same set of the partition.
1817:
1797:
1771:
1751:
1731:
1711:
1687:
1657:
1631:
1602:
1582:
1562:
1518:
1514:
1509:
1482:
1448:
1437:
1408:
1326:
1321:
1290:
1285:
1232:
1227:
1207:
1175:
1152:
1132:
1104:
1082:
1056:
1051:
1031:
1011:
926:
903:
857:
831:
805:
779:
740:
714:
688:
656:
630:
607:
591:
587:
585:
562:
506:
502:
500:
472:
447:
442:
437:
418:
414:
412:
392:
361:
334:
313:
301:
275:
255:
235:
208:
192:
188:
186:
166:
136:
116:
92:
68:
4352:An Introduction to Mathematical Thinking
2750:
1026:with respect to an equivalence relation
87:), then one may naturally split the set
43:
4096:
3950:, where the quotient homomorphism is a
3161:A frequent particular case occurs when
2790:where the vertices are the elements of
2654:consists of all the lines in, say, the
2557:then the equivalence class of the pair
4325:Analysis with an introduction to proof
4180:
4144:
4120:
4103:
4038:of equivalence relations given above.
1395:are equivalent—in this case, one says
1006:The set of all equivalence classes in
999:, and the equivalence classes, called
329:to emphasize its equivalence relation
4288:(6th ed.), Thomson (Brooks/Cole)
4050:, a method for devising test sets in
3946:is a vector space formed by taking a
2730:line is considered parallel to itself
2342:{\displaystyle \mathbb {Z} /{\sim }.}
2162:2 equivalence relation on the set of
2052:{\displaystyle \cap \neq \emptyset .}
1597:is a member of the equivalence class
7:
4409:Introduction to Advanced Mathematics
4400:Introduction to Abstract Mathematics
4286:A Transition to Advanced Mathematics
4198:Foundations for Advanced Mathematics
4132:
1857:is an equivalence relation on a set
973:{\displaystyle =\{x\in X:a\sim x\}.}
898:The equivalence class of an element
4073:Quotient by an equivalence relation
2430:and define an equivalence relation
1456:
1449:
2314:all represent the same element of
2043:
1537:and produces the remainder of the
1470:{\textstyle a\equiv b{\pmod {m}}.}
25:
4361:Foundations of Higher Mathematics
4234:Mathematical Thinking and Writing
3401:This occurs, for example, in the
622:satisfying the three properties:
519:quotient spaces in linear algebra
4497:
4431:
4418:A Primer of Abstract Mathematics
3319:{\displaystyle x_{1}\sim x_{2},}
2217:if and only if their difference
1351:, when using the terminology of
995:" is an equivalence relation on
4284:Smith; Eggen; St.Andre (2006),
4035:
3570:{\displaystyle x_{1}\sim x_{2}}
2515:{\displaystyle (a,b)\sim (c,d)}
495:) and the equivalence relation
4334:Bridge to Abstract Mathematics
3760:
3754:
3727:
3721:
3698:
3692:
3662:is the set of all elements in
3493:
3449:" instead of "invariant under
3069:
3063:
3040:
3034:
2963:
2957:
2924:is an equivalence relation on
2576:
2564:
2509:
2497:
2491:
2479:
2394:
2382:
2301:
2295:
2272:
2266:
2260:
2254:
2037:
2031:
2025:
2019:
1998:
1992:
1986:
1980:
1665:
1659:
1639:
1633:
1626:Every two equivalence classes
1610:
1604:
1460:
1450:
1347:such an injection is called a
1189:
1183:
1180:
934:
928:
487:has some structure (such as a
310:
303:
283:
277:
1:
2976:is a property of elements of
2184:{\displaystyle \mathbb {Z} ,}
1530:{\displaystyle a{\bmod {m}},}
4391:The Nuts and Bolts of Proofs
4236:, Harcourt/ Academic Press,
4067:Partial equivalence relation
181:and an equivalence relation
60:, when the elements of some
4232:Maddox, Randall B. (2002),
4085:Transversal (combinatorics)
3082:is true, then the property
2589:can be identified with the
523:quotient spaces in topology
5138:
4957:von Neumann–Bernays–Gödel
4350:Gilbert; Vanstone (2005),
4332:Morash, Ronald P. (1987),
4196:Avelsgaard, Carol (1989),
3907:Quotient space in topology
3530:{\displaystyle \,\sim \,,}
3367:{\displaystyle \,\sim \,,}
3124:{\displaystyle \,\sim \,,}
2853:are joined if and only if
2744:
883:{\displaystyle a,b,c\in X}
458:{\displaystyle S/{\sim }.}
425:{\displaystyle \,\sim \,,}
36:
29:
5117:Equivalence (mathematics)
4758:One-to-one correspondence
4495:
4407:Barnier; Feldman (2000),
3872:{\displaystyle \sim _{Y}}
3825:{\displaystyle \sim _{X}}
3642:The equivalence class of
3464:{\displaystyle \,\sim \,}
3442:{\displaystyle \,\sim \,}
3420:{\displaystyle \,\sim \,}
2917:{\displaystyle \,\sim \,}
2763:may be associated to any
2445:{\displaystyle \,\sim \,}
2103:{\displaystyle \,\sim \,}
1850:{\displaystyle \,\sim \,}
1195:{\displaystyle x\mapsto }
595:{\displaystyle \,\sim \,}
510:{\displaystyle \,\sim \,}
196:{\displaystyle \,\sim \,}
32:equivalence class (music)
4250:Wolf, Robert S. (1998),
4060:, the quotient space of
4048:Equivalence partitioning
3899:). Such a function is a
3502:{\displaystyle f:X\to Y}
3394:{\displaystyle \,\sim .}
3152:{\displaystyle \,\sim .}
3018:{\displaystyle x\sim y,}
2875:{\displaystyle s\sim t.}
2741:Graphical representation
760:{\displaystyle a,b\in X}
322:{\displaystyle _{\sim }}
4377:D'Angelo; West (2000),
4354:, Pearson Prentice-Hall
2677:{\displaystyle L\sim M}
2210:{\displaystyle x\sim y}
1965:{\displaystyle x\sim y}
1785:{\displaystyle x\sim y}
845:{\displaystyle a\sim c}
819:{\displaystyle b\sim c}
793:{\displaystyle a\sim b}
728:{\displaystyle b\sim a}
702:{\displaystyle a\sim b}
644:{\displaystyle a\sim a}
549:Definition and notation
158:, they are equivalent.
39:Quotient map (topology)
4716:Constructible universe
4543:Constructibility (V=L)
4214:Devlin, Keith (2004),
4024:
4001:
3893:
3873:
3846:
3826:
3797:
3770:
3734:
3708:
3676:
3656:
3636:
3571:
3531:
3503:
3465:
3443:
3421:
3395:
3368:
3340:
3320:
3277:
3215:
3195:
3175:
3153:
3125:
3096:
3076:
3047:
3019:
2990:
2970:
2941:
2918:
2876:
2847:
2827:
2807:
2784:
2756:
2718:
2698:
2678:
2648:
2617:
2583:
2551:
2550:{\displaystyle ad=bc,}
2516:
2466:
2446:
2424:
2401:
2365:
2343:
2308:
2282:
2237:
2211:
2185:
2150:
2127:
2104:
2082:
2053:
2005:
1966:
1937:
1914:
1894:
1874:
1851:
1826:
1806:
1786:
1760:
1740:
1720:
1696:
1672:
1646:
1620:
1591:
1571:
1531:
1494:
1471:
1426:
1341:
1302:
1247:
1216:
1196:
1161:
1141:
1113:
1091:
1071:
1040:
1020:
991:. For example, "being
974:
912:
884:
846:
820:
794:
761:
729:
703:
671:
670:{\displaystyle a\in X}
645:
616:
596:
571:
511:
481:
459:
426:
401:
373:
346:
345:{\displaystyle \sim .}
323:
290:
264:
244:
220:
197:
175:
161:Formally, given a set
145:
125:
101:
77:
53:
4939:Principia Mathematica
4773:Transfinite induction
4632:(i.e. set difference)
4025:
4002:
3894:
3874:
3847:
3827:
3798:
3771:
3769:{\displaystyle f(x).}
3735:
3709:
3707:{\displaystyle f(x),}
3677:
3657:
3637:
3572:
3532:
3511:class invariant under
3504:
3466:
3444:
3422:
3396:
3369:
3348:class invariant under
3341:
3321:
3278:
3216:
3196:
3176:
3154:
3126:
3097:
3077:
3048:
3020:
2991:
2971:
2942:
2919:
2877:
2848:
2828:
2808:
2785:
2754:
2719:
2699:
2679:
2649:
2618:
2584:
2582:{\displaystyle (a,b)}
2552:
2517:
2467:
2447:
2425:
2402:
2400:{\displaystyle (a,b)}
2366:
2344:
2309:
2283:
2238:
2212:
2186:
2151:
2128:
2105:
2083:
2054:
2006:
1967:
1938:
1915:
1895:
1875:
1852:
1827:
1807:
1787:
1761:
1741:
1721:
1697:
1673:
1647:
1621:
1592:
1572:
1532:
1495:
1472:
1427:
1342:
1303:
1248:
1217:
1197:
1162:
1142:
1114:
1092:
1072:
1041:
1021:
975:
913:
885:
847:
821:
795:
762:
730:
704:
672:
646:
617:
597:
572:
512:
482:
460:
427:
402:
374:
347:
324:
291:
265:
245:
221:
198:
176:
146:
126:
102:
78:
47:
5013:Burali-Forti paradox
4768:Set-builder notation
4721:Continuum hypothesis
4661:Symmetric difference
4440:at Wikimedia Commons
4147:, p. 74, Thm. 2.5.15
4011:
3988:
3932:congruence relations
3883:
3856:
3836:
3809:
3784:
3748:
3718:
3686:
3682:which get mapped to
3666:
3646:
3581:
3541:
3516:
3481:
3453:
3431:
3427:" or just "respects
3409:
3381:
3353:
3330:
3287:
3225:
3205:
3185:
3165:
3139:
3110:
3086:
3075:{\displaystyle P(y)}
3057:
3046:{\displaystyle P(x)}
3028:
3000:
2980:
2969:{\displaystyle P(x)}
2951:
2928:
2906:
2888:connected components
2857:
2837:
2817:
2794:
2771:
2708:
2688:
2662:
2638:
2616:{\displaystyle a/b,}
2596:
2561:
2526:
2476:
2456:
2434:
2411:
2379:
2355:
2318:
2292:
2251:
2221:
2195:
2170:
2137:
2114:
2092:
2072:
2016:
1977:
1950:
1924:
1920:are two elements of
1904:
1884:
1861:
1839:
1816:
1796:
1770:
1750:
1730:
1710:
1686:
1678:are either equal or
1656:
1630:
1601:
1581:
1561:
1508:
1504:function is denoted
1481:
1436:
1425:{\displaystyle a-b;}
1407:
1340:{\displaystyle X/R,}
1320:
1284:
1263:canonical projection
1257:canonical surjection
1246:{\displaystyle X/R,}
1226:
1206:
1174:
1151:
1131:
1103:
1081:
1070:{\displaystyle X/R,}
1050:
1030:
1010:
925:
902:
856:
830:
804:
778:
739:
713:
687:
655:
629:
606:
584:
561:
555:equivalence relation
499:
471:
436:
411:
391:
360:
333:
300:
274:
254:
234:
207:
185:
165:
135:
115:
91:
85:equivalence relation
67:
27:Mathematical concept
4974:Tarski–Grothendieck
4438:Equivalence classes
4159:, p. 132, Thm. 3.16
3714:that is, the class
3537:according to which
3181:is a function from
3135:under the relation
2996:such that whenever
2236:{\displaystyle x-y}
1726:: every element of
1301:{\displaystyle X/R}
1001:isomorphism classes
543:quotient categories
151:belong to the same
109:equivalence classes
4563:Limitation of size
4368:Iglewicz; Stoyle,
4300:, Addison-Wesley,
4275:Sundstrom (2003),
4200:, Scott Foresman,
4023:{\displaystyle X,}
4020:
4000:{\displaystyle X,}
3997:
3889:
3869:
3842:
3822:
3796:{\displaystyle f.}
3793:
3766:
3730:
3704:
3672:
3652:
3632:
3567:
3527:
3499:
3461:
3439:
3417:
3391:
3364:
3336:
3316:
3273:
3211:
3191:
3171:
3149:
3121:
3092:
3072:
3043:
3015:
2986:
2966:
2940:{\displaystyle X,}
2937:
2914:
2872:
2843:
2823:
2806:{\displaystyle X,}
2803:
2783:{\displaystyle X,}
2780:
2765:symmetric relation
2757:
2714:
2694:
2674:
2644:
2625:field of fractions
2613:
2579:
2547:
2512:
2462:
2442:
2423:{\displaystyle b,}
2420:
2397:
2361:
2339:
2304:
2281:{\displaystyle ,,}
2278:
2233:
2207:
2181:
2149:{\displaystyle A.}
2146:
2126:{\displaystyle A,}
2123:
2100:
2078:
2049:
2001:
1962:
1936:{\displaystyle X,}
1933:
1910:
1890:
1873:{\displaystyle X,}
1870:
1847:
1822:
1802:
1782:
1756:
1736:
1716:
1692:
1668:
1642:
1616:
1587:
1567:
1539:Euclidean division
1527:
1493:{\displaystyle m,}
1490:
1467:
1422:
1382:congruence modulo
1367:modular arithmetic
1365:. For example, in
1337:
1298:
1243:
1212:
1192:
1157:
1137:
1109:
1087:
1067:
1036:
1016:
970:
908:
880:
842:
816:
790:
757:
725:
699:
667:
641:
612:
592:
567:
531:homogeneous spaces
507:
477:
455:
432:and is denoted by
422:
397:
372:{\displaystyle S,}
369:
342:
319:
296:or, equivalently,
286:
260:
240:
219:{\displaystyle S,}
216:
193:
171:
141:
121:
97:
73:
54:
52:equivalence class.
5094:
5093:
5003:Russell's paradox
4952:Zermelo–Fraenkel
4853:Dedekind-infinite
4726:Diagonal argument
4625:Cartesian product
4489:Set (mathematics)
4436:Media related to
4359:Fletcher; Patty,
4294:Schumacher, Carol
4261:978-0-7167-3050-7
4225:978-1-58488-449-1
4058:Homogeneous space
3921:topological space
3892:{\displaystyle Y}
3845:{\displaystyle X}
3675:{\displaystyle X}
3655:{\displaystyle x}
3339:{\displaystyle f}
3214:{\displaystyle Y}
3194:{\displaystyle X}
3174:{\displaystyle f}
3102:is said to be an
3095:{\displaystyle P}
2989:{\displaystyle X}
2846:{\displaystyle t}
2826:{\displaystyle s}
2813:and two vertices
2734:point at infinity
2717:{\displaystyle M}
2697:{\displaystyle L}
2647:{\displaystyle X}
2465:{\displaystyle X}
2364:{\displaystyle X}
2081:{\displaystyle X}
2004:{\displaystyle =}
1913:{\displaystyle y}
1893:{\displaystyle x}
1825:{\displaystyle y}
1805:{\displaystyle x}
1759:{\displaystyle X}
1739:{\displaystyle X}
1719:{\displaystyle X}
1695:{\displaystyle X}
1619:{\displaystyle .}
1590:{\displaystyle X}
1570:{\displaystyle x}
1215:{\displaystyle X}
1160:{\displaystyle R}
1140:{\displaystyle X}
1112:{\displaystyle R}
1090:{\displaystyle X}
1039:{\displaystyle R}
1019:{\displaystyle X}
911:{\displaystyle a}
615:{\displaystyle X}
570:{\displaystyle X}
480:{\displaystyle S}
400:{\displaystyle S}
263:{\displaystyle S}
243:{\displaystyle a}
228:equivalence class
174:{\displaystyle S}
153:equivalence class
144:{\displaystyle b}
124:{\displaystyle a}
100:{\displaystyle S}
76:{\displaystyle S}
16:(Redirected from
5129:
5112:Binary relations
5076:Bertrand Russell
5066:John von Neumann
5051:Abraham Fraenkel
5046:Richard Dedekind
5008:Suslin's problem
4919:Cantor's theorem
4636:De Morgan's laws
4501:
4468:
4461:
4454:
4445:
4435:
4421:
4412:
4403:
4394:
4382:
4373:
4364:
4355:
4346:
4336:, Random House,
4328:
4319:
4314:O'Leary (2003),
4310:
4289:
4280:
4264:
4246:
4228:
4210:
4184:
4178:
4172:
4166:
4160:
4154:
4148:
4142:
4136:
4130:
4124:
4118:
4107:
4101:
4052:software testing
4029:
4027:
4026:
4021:
4006:
4004:
4003:
3998:
3969:The orbits of a
3956:quotient modules
3936:quotient algebra
3928:abstract algebra
3898:
3896:
3895:
3890:
3878:
3876:
3875:
3870:
3868:
3867:
3851:
3849:
3848:
3843:
3831:
3829:
3828:
3823:
3821:
3820:
3802:
3800:
3799:
3794:
3775:
3773:
3772:
3767:
3739:
3737:
3736:
3733:{\displaystyle }
3731:
3713:
3711:
3710:
3705:
3681:
3679:
3678:
3673:
3661:
3659:
3658:
3653:
3641:
3639:
3638:
3633:
3628:
3624:
3623:
3604:
3600:
3599:
3576:
3574:
3573:
3568:
3566:
3565:
3553:
3552:
3536:
3534:
3533:
3528:
3508:
3506:
3505:
3500:
3470:
3468:
3467:
3462:
3448:
3446:
3445:
3440:
3426:
3424:
3423:
3418:
3403:character theory
3400:
3398:
3397:
3392:
3373:
3371:
3370:
3365:
3345:
3343:
3342:
3337:
3325:
3323:
3322:
3317:
3312:
3311:
3299:
3298:
3282:
3280:
3279:
3274:
3272:
3268:
3267:
3248:
3244:
3243:
3220:
3218:
3217:
3212:
3200:
3198:
3197:
3192:
3180:
3178:
3177:
3172:
3158:
3156:
3155:
3150:
3130:
3128:
3127:
3122:
3101:
3099:
3098:
3093:
3081:
3079:
3078:
3073:
3052:
3050:
3049:
3044:
3024:
3022:
3021:
3016:
2995:
2993:
2992:
2987:
2975:
2973:
2972:
2967:
2946:
2944:
2943:
2938:
2923:
2921:
2920:
2915:
2881:
2879:
2878:
2873:
2852:
2850:
2849:
2844:
2832:
2830:
2829:
2824:
2812:
2810:
2809:
2804:
2789:
2787:
2786:
2781:
2761:undirected graph
2723:
2721:
2720:
2715:
2703:
2701:
2700:
2695:
2683:
2681:
2680:
2675:
2653:
2651:
2650:
2645:
2622:
2620:
2619:
2614:
2606:
2588:
2586:
2585:
2580:
2556:
2554:
2553:
2548:
2521:
2519:
2518:
2513:
2471:
2469:
2468:
2463:
2451:
2449:
2448:
2443:
2429:
2427:
2426:
2421:
2406:
2404:
2403:
2398:
2370:
2368:
2367:
2362:
2348:
2346:
2345:
2340:
2335:
2330:
2325:
2313:
2311:
2310:
2307:{\displaystyle }
2305:
2287:
2285:
2284:
2279:
2242:
2240:
2239:
2234:
2216:
2214:
2213:
2208:
2190:
2188:
2187:
2182:
2177:
2155:
2153:
2152:
2147:
2132:
2130:
2129:
2124:
2109:
2107:
2106:
2101:
2087:
2085:
2084:
2079:
2058:
2056:
2055:
2050:
2010:
2008:
2007:
2002:
1971:
1969:
1968:
1963:
1942:
1940:
1939:
1934:
1919:
1917:
1916:
1911:
1899:
1897:
1896:
1891:
1879:
1877:
1876:
1871:
1856:
1854:
1853:
1848:
1831:
1829:
1828:
1823:
1811:
1809:
1808:
1803:
1791:
1789:
1788:
1783:
1765:
1763:
1762:
1757:
1745:
1743:
1742:
1737:
1725:
1723:
1722:
1717:
1701:
1699:
1698:
1693:
1677:
1675:
1674:
1671:{\displaystyle }
1669:
1651:
1649:
1648:
1645:{\displaystyle }
1643:
1625:
1623:
1622:
1617:
1596:
1594:
1593:
1588:
1576:
1574:
1573:
1568:
1548:
1544:
1536:
1534:
1533:
1528:
1523:
1522:
1499:
1497:
1496:
1491:
1476:
1474:
1473:
1468:
1463:
1432:this is denoted
1431:
1429:
1428:
1423:
1402:
1394:
1390:
1385:
1379:
1375:
1346:
1344:
1343:
1338:
1330:
1311:
1307:
1305:
1304:
1299:
1294:
1259:
1258:
1252:
1250:
1249:
1244:
1236:
1221:
1219:
1218:
1213:
1201:
1199:
1198:
1193:
1166:
1164:
1163:
1158:
1146:
1144:
1143:
1138:
1125:
1124:
1118:
1116:
1115:
1110:
1096:
1094:
1093:
1088:
1076:
1074:
1073:
1068:
1060:
1045:
1043:
1042:
1037:
1025:
1023:
1022:
1017:
1003:, are not sets.
979:
977:
976:
971:
917:
915:
914:
909:
889:
887:
886:
881:
851:
849:
848:
843:
825:
823:
822:
817:
799:
797:
796:
791:
766:
764:
763:
758:
734:
732:
731:
726:
708:
706:
705:
700:
676:
674:
673:
668:
650:
648:
647:
642:
621:
619:
618:
613:
601:
599:
598:
593:
576:
574:
573:
568:
539:quotient monoids
516:
514:
513:
508:
486:
484:
483:
478:
464:
462:
461:
456:
451:
446:
431:
429:
428:
423:
406:
404:
403:
398:
378:
376:
375:
370:
351:
349:
348:
343:
328:
326:
325:
320:
318:
317:
295:
293:
292:
289:{\displaystyle }
287:
269:
267:
266:
261:
249:
247:
246:
241:
225:
223:
222:
217:
202:
200:
199:
194:
180:
178:
177:
172:
150:
148:
147:
142:
130:
128:
127:
122:
106:
104:
103:
98:
82:
80:
79:
74:
21:
5137:
5136:
5132:
5131:
5130:
5128:
5127:
5126:
5097:
5096:
5095:
5090:
5017:
4996:
4980:
4945:New Foundations
4892:
4782:
4701:Cardinal number
4684:
4670:
4611:
4502:
4493:
4477:
4472:
4428:
4415:
4411:, Prentice Hall
4406:
4397:
4385:
4381:, Prentice Hall
4376:
4367:
4358:
4349:
4344:
4331:
4327:, Prentice Hall
4322:
4318:, Prentice-Hall
4313:
4308:
4292:
4283:
4279:, Prentice-Hall
4274:
4271:
4269:Further reading
4262:
4249:
4244:
4231:
4226:
4213:
4208:
4195:
4192:
4187:
4179:
4175:
4169:Avelsgaard 1989
4167:
4163:
4157:Avelsgaard 1989
4155:
4151:
4143:
4139:
4131:
4127:
4119:
4110:
4102:
4098:
4094:
4044:
4009:
4008:
3986:
3985:
3964:quotient groups
3909:
3881:
3880:
3859:
3854:
3853:
3834:
3833:
3812:
3807:
3806:
3782:
3781:
3746:
3745:
3716:
3715:
3684:
3683:
3664:
3663:
3644:
3643:
3615:
3611:
3591:
3587:
3579:
3578:
3577:if and only if
3557:
3544:
3539:
3538:
3514:
3513:
3479:
3478:
3451:
3450:
3429:
3428:
3407:
3406:
3379:
3378:
3376:invariant under
3351:
3350:
3328:
3327:
3303:
3290:
3285:
3284:
3259:
3255:
3235:
3231:
3223:
3222:
3203:
3202:
3201:to another set
3183:
3182:
3163:
3162:
3137:
3136:
3108:
3107:
3084:
3083:
3055:
3054:
3026:
3025:
2998:
2997:
2978:
2977:
2949:
2948:
2926:
2925:
2904:
2903:
2900:
2855:
2854:
2835:
2834:
2815:
2814:
2792:
2791:
2769:
2768:
2749:
2743:
2706:
2705:
2686:
2685:
2660:
2659:
2656:Euclidean plane
2636:
2635:
2629:integral domain
2594:
2593:
2591:rational number
2559:
2558:
2524:
2523:
2522:if and only if
2474:
2473:
2454:
2453:
2432:
2431:
2409:
2408:
2377:
2376:
2353:
2352:
2316:
2315:
2290:
2289:
2249:
2248:
2219:
2218:
2193:
2192:
2168:
2167:
2135:
2134:
2112:
2111:
2090:
2089:
2070:
2069:
2065:
2014:
2013:
1975:
1974:
1948:
1947:
1922:
1921:
1902:
1901:
1882:
1881:
1859:
1858:
1837:
1836:
1814:
1813:
1794:
1793:
1792:if and only if
1768:
1767:
1748:
1747:
1728:
1727:
1708:
1707:
1684:
1683:
1654:
1653:
1628:
1627:
1599:
1598:
1579:
1578:
1559:
1558:
1555:
1546:
1542:
1506:
1505:
1479:
1478:
1434:
1433:
1405:
1404:
1400:
1392:
1388:
1383:
1377:
1373:
1363:representatives
1353:category theory
1318:
1317:
1309:
1282:
1281:
1256:
1255:
1224:
1223:
1204:
1203:
1172:
1171:
1149:
1148:
1129:
1128:
1122:
1121:
1101:
1100:
1079:
1078:
1048:
1047:
1028:
1027:
1008:
1007:
923:
922:
900:
899:
854:
853:
828:
827:
802:
801:
776:
775:
737:
736:
711:
710:
685:
684:
653:
652:
627:
626:
604:
603:
582:
581:
579:binary relation
559:
558:
551:
527:quotient groups
497:
496:
489:group operation
469:
468:
434:
433:
409:
408:
389:
388:
358:
357:
331:
330:
309:
298:
297:
272:
271:
252:
251:
232:
231:
205:
204:
183:
182:
163:
162:
156:if, and only if
133:
132:
113:
112:
89:
88:
65:
64:
42:
35:
28:
23:
22:
15:
12:
11:
5:
5135:
5133:
5125:
5124:
5119:
5114:
5109:
5099:
5098:
5092:
5091:
5089:
5088:
5083:
5081:Thoralf Skolem
5078:
5073:
5068:
5063:
5058:
5053:
5048:
5043:
5038:
5033:
5027:
5025:
5019:
5018:
5016:
5015:
5010:
5005:
4999:
4997:
4995:
4994:
4991:
4985:
4982:
4981:
4979:
4978:
4977:
4976:
4971:
4966:
4965:
4964:
4949:
4948:
4947:
4935:
4934:
4933:
4922:
4921:
4916:
4911:
4906:
4900:
4898:
4894:
4893:
4891:
4890:
4885:
4880:
4875:
4866:
4861:
4856:
4846:
4841:
4840:
4839:
4834:
4829:
4819:
4809:
4804:
4799:
4793:
4791:
4784:
4783:
4781:
4780:
4775:
4770:
4765:
4763:Ordinal number
4760:
4755:
4750:
4745:
4744:
4743:
4738:
4728:
4723:
4718:
4713:
4708:
4698:
4693:
4687:
4685:
4683:
4682:
4679:
4675:
4672:
4671:
4669:
4668:
4663:
4658:
4653:
4648:
4643:
4641:Disjoint union
4638:
4633:
4627:
4621:
4619:
4613:
4612:
4610:
4609:
4608:
4607:
4602:
4591:
4590:
4588:Martin's axiom
4585:
4580:
4575:
4570:
4565:
4560:
4555:
4553:Extensionality
4550:
4545:
4540:
4539:
4538:
4533:
4528:
4518:
4512:
4510:
4504:
4503:
4496:
4494:
4492:
4491:
4485:
4483:
4479:
4478:
4473:
4471:
4470:
4463:
4456:
4448:
4442:
4441:
4427:
4426:External links
4424:
4423:
4422:
4413:
4404:
4395:
4383:
4374:
4365:
4356:
4347:
4342:
4329:
4320:
4311:
4306:
4290:
4281:
4270:
4267:
4266:
4265:
4260:
4247:
4242:
4229:
4224:
4211:
4206:
4191:
4188:
4186:
4185:
4173:
4161:
4149:
4137:
4125:
4123:, p. 123.
4108:
4106:, p. 122.
4095:
4093:
4090:
4089:
4088:
4082:
4076:
4070:
4064:
4055:
4043:
4040:
4019:
4016:
3996:
3993:
3960:quotient rings
3948:quotient group
3944:quotient space
3940:linear algebra
3917:quotient space
3908:
3905:
3888:
3866:
3862:
3841:
3819:
3815:
3792:
3789:
3765:
3762:
3759:
3756:
3753:
3729:
3726:
3723:
3703:
3700:
3697:
3694:
3691:
3671:
3651:
3631:
3627:
3622:
3618:
3614:
3610:
3607:
3603:
3598:
3594:
3590:
3586:
3564:
3560:
3556:
3551:
3547:
3526:
3522:
3498:
3495:
3492:
3489:
3486:
3459:
3437:
3415:
3390:
3387:
3377:
3363:
3359:
3349:
3346:is said to be
3335:
3315:
3310:
3306:
3302:
3297:
3293:
3271:
3266:
3262:
3258:
3254:
3251:
3247:
3242:
3238:
3234:
3230:
3210:
3190:
3170:
3148:
3145:
3120:
3116:
3091:
3071:
3068:
3065:
3062:
3042:
3039:
3036:
3033:
3014:
3011:
3008:
3005:
2985:
2965:
2962:
2959:
2956:
2936:
2933:
2912:
2899:
2896:
2884:cluster graphs
2871:
2868:
2865:
2862:
2842:
2822:
2802:
2799:
2779:
2776:
2745:Main article:
2742:
2739:
2738:
2737:
2726:parallel lines
2713:
2693:
2673:
2670:
2667:
2643:
2632:
2612:
2609:
2605:
2601:
2578:
2575:
2572:
2569:
2566:
2546:
2543:
2540:
2537:
2534:
2531:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2461:
2440:
2419:
2416:
2407:with non-zero
2396:
2393:
2390:
2387:
2384:
2371:be the set of
2360:
2349:
2338:
2334:
2329:
2324:
2303:
2300:
2297:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2232:
2229:
2226:
2206:
2203:
2200:
2180:
2176:
2156:
2145:
2142:
2122:
2119:
2098:
2077:
2064:
2061:
2060:
2059:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2011:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1972:
1961:
1958:
1955:
1932:
1929:
1909:
1889:
1869:
1866:
1845:
1821:
1801:
1781:
1778:
1775:
1755:
1735:
1715:
1691:
1667:
1664:
1661:
1641:
1638:
1635:
1615:
1612:
1609:
1606:
1586:
1566:
1557:Every element
1554:
1551:
1526:
1521:
1517:
1513:
1489:
1486:
1466:
1462:
1459:
1455:
1452:
1447:
1444:
1441:
1421:
1418:
1415:
1412:
1364:
1336:
1333:
1329:
1325:
1297:
1293:
1289:
1274:representative
1242:
1239:
1235:
1231:
1211:
1191:
1188:
1185:
1182:
1179:
1169:surjective map
1156:
1136:
1108:
1086:
1077:and is called
1066:
1063:
1059:
1055:
1046:is denoted as
1035:
1015:
989:proper classes
981:
980:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
918:is defined as
907:
896:
895:
879:
876:
873:
870:
867:
864:
861:
841:
838:
835:
815:
812:
809:
789:
786:
783:
772:
756:
753:
750:
747:
744:
724:
721:
718:
698:
695:
692:
682:
666:
663:
660:
640:
637:
634:
611:
590:
566:
550:
547:
535:quotient rings
505:
476:
454:
450:
445:
441:
421:
417:
396:
385:quotient space
368:
365:
341:
338:
316:
312:
308:
305:
285:
282:
279:
259:
239:
230:of an element
229:
215:
212:
191:
170:
140:
120:
96:
72:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5134:
5123:
5120:
5118:
5115:
5113:
5110:
5108:
5105:
5104:
5102:
5087:
5086:Ernst Zermelo
5084:
5082:
5079:
5077:
5074:
5072:
5071:Willard Quine
5069:
5067:
5064:
5062:
5059:
5057:
5054:
5052:
5049:
5047:
5044:
5042:
5039:
5037:
5034:
5032:
5029:
5028:
5026:
5024:
5023:Set theorists
5020:
5014:
5011:
5009:
5006:
5004:
5001:
5000:
4998:
4992:
4990:
4987:
4986:
4983:
4975:
4972:
4970:
4969:Kripke–Platek
4967:
4963:
4960:
4959:
4958:
4955:
4954:
4953:
4950:
4946:
4943:
4942:
4941:
4940:
4936:
4932:
4929:
4928:
4927:
4924:
4923:
4920:
4917:
4915:
4912:
4910:
4907:
4905:
4902:
4901:
4899:
4895:
4889:
4886:
4884:
4881:
4879:
4876:
4874:
4872:
4867:
4865:
4862:
4860:
4857:
4854:
4850:
4847:
4845:
4842:
4838:
4835:
4833:
4830:
4828:
4825:
4824:
4823:
4820:
4817:
4813:
4810:
4808:
4805:
4803:
4800:
4798:
4795:
4794:
4792:
4789:
4785:
4779:
4776:
4774:
4771:
4769:
4766:
4764:
4761:
4759:
4756:
4754:
4751:
4749:
4746:
4742:
4739:
4737:
4734:
4733:
4732:
4729:
4727:
4724:
4722:
4719:
4717:
4714:
4712:
4709:
4706:
4702:
4699:
4697:
4694:
4692:
4689:
4688:
4686:
4680:
4677:
4676:
4673:
4667:
4664:
4662:
4659:
4657:
4654:
4652:
4649:
4647:
4644:
4642:
4639:
4637:
4634:
4631:
4628:
4626:
4623:
4622:
4620:
4618:
4614:
4606:
4605:specification
4603:
4601:
4598:
4597:
4596:
4593:
4592:
4589:
4586:
4584:
4581:
4579:
4576:
4574:
4571:
4569:
4566:
4564:
4561:
4559:
4556:
4554:
4551:
4549:
4546:
4544:
4541:
4537:
4534:
4532:
4529:
4527:
4524:
4523:
4522:
4519:
4517:
4514:
4513:
4511:
4509:
4505:
4500:
4490:
4487:
4486:
4484:
4480:
4476:
4469:
4464:
4462:
4457:
4455:
4450:
4449:
4446:
4439:
4434:
4430:
4429:
4425:
4419:
4414:
4410:
4405:
4402:, Brooks/Cole
4401:
4396:
4392:
4388:
4384:
4380:
4375:
4371:
4366:
4362:
4357:
4353:
4348:
4345:
4343:0-394-35429-X
4339:
4335:
4330:
4326:
4321:
4317:
4312:
4309:
4307:0-201-82653-4
4303:
4299:
4295:
4291:
4287:
4282:
4278:
4273:
4272:
4268:
4263:
4257:
4253:
4248:
4245:
4243:0-12-464976-9
4239:
4235:
4230:
4227:
4221:
4217:
4212:
4209:
4207:0-673-38152-8
4203:
4199:
4194:
4193:
4189:
4182:
4177:
4174:
4170:
4165:
4162:
4158:
4153:
4150:
4146:
4141:
4138:
4134:
4129:
4126:
4122:
4117:
4115:
4113:
4109:
4105:
4100:
4097:
4091:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4063:
4059:
4056:
4053:
4049:
4046:
4045:
4041:
4039:
4037:
4033:
4017:
4014:
3994:
3991:
3981:
3978:
3976:
3972:
3967:
3965:
3961:
3957:
3953:
3949:
3945:
3941:
3937:
3933:
3929:
3924:
3922:
3918:
3914:
3906:
3904:
3902:
3886:
3864:
3860:
3839:
3817:
3813:
3803:
3790:
3787:
3779:
3763:
3757:
3751:
3743:
3742:inverse image
3724:
3701:
3695:
3689:
3669:
3649:
3629:
3625:
3620:
3616:
3612:
3608:
3605:
3601:
3596:
3592:
3588:
3584:
3562:
3558:
3554:
3549:
3545:
3524:
3520:
3512:
3496:
3490:
3487:
3484:
3477:
3472:
3457:
3435:
3413:
3404:
3388:
3385:
3375:
3361:
3357:
3347:
3333:
3313:
3308:
3304:
3300:
3295:
3291:
3269:
3264:
3260:
3256:
3252:
3249:
3245:
3240:
3236:
3232:
3228:
3208:
3188:
3168:
3159:
3146:
3143:
3134:
3118:
3114:
3105:
3089:
3066:
3060:
3037:
3031:
3012:
3009:
3006:
3003:
2983:
2960:
2954:
2934:
2931:
2910:
2897:
2895:
2893:
2889:
2885:
2869:
2866:
2863:
2860:
2840:
2820:
2800:
2797:
2777:
2774:
2766:
2762:
2753:
2748:
2747:Cluster graph
2740:
2735:
2731:
2727:
2711:
2691:
2671:
2668:
2665:
2657:
2641:
2633:
2630:
2626:
2610:
2607:
2603:
2599:
2592:
2573:
2570:
2567:
2544:
2541:
2538:
2535:
2532:
2529:
2506:
2503:
2500:
2494:
2488:
2485:
2482:
2459:
2438:
2417:
2414:
2391:
2388:
2385:
2374:
2373:ordered pairs
2358:
2350:
2336:
2332:
2327:
2298:
2275:
2269:
2263:
2257:
2246:
2230:
2227:
2224:
2204:
2201:
2198:
2178:
2165:
2161:
2158:Consider the
2157:
2143:
2140:
2120:
2117:
2096:
2075:
2067:
2066:
2062:
2046:
2040:
2034:
2028:
2022:
2012:
1995:
1989:
1983:
1973:
1959:
1956:
1953:
1946:
1945:
1944:
1930:
1927:
1907:
1887:
1867:
1864:
1843:
1833:
1819:
1799:
1779:
1776:
1773:
1753:
1733:
1713:
1705:
1689:
1681:
1662:
1636:
1613:
1607:
1584:
1564:
1552:
1550:
1540:
1524:
1519:
1511:
1501:
1487:
1484:
1464:
1457:
1453:
1445:
1442:
1439:
1419:
1416:
1413:
1410:
1398:
1386:
1376:greater than
1372:
1368:
1362:
1359:
1356:
1354:
1350:
1334:
1331:
1327:
1323:
1315:
1295:
1291:
1287:
1279:
1275:
1271:
1266:
1264:
1260:
1240:
1237:
1233:
1229:
1209:
1186:
1177:
1170:
1154:
1134:
1126:
1106:
1099:
1084:
1064:
1061:
1057:
1053:
1033:
1013:
1004:
1002:
998:
994:
990:
986:
967:
961:
958:
955:
952:
949:
946:
943:
937:
931:
921:
920:
919:
905:
893:
877:
874:
871:
868:
865:
862:
859:
839:
836:
833:
813:
810:
807:
787:
784:
781:
773:
770:
754:
751:
748:
745:
742:
722:
719:
716:
696:
693:
690:
683:
680:
664:
661:
658:
638:
635:
632:
625:
624:
623:
609:
588:
580:
564:
556:
548:
546:
544:
540:
536:
532:
528:
524:
520:
503:
494:
490:
474:
467:When the set
465:
452:
448:
443:
439:
419:
415:
394:
386:
382:
366:
363:
355:
339:
336:
314:
306:
280:
257:
237:
227:
213:
210:
189:
168:
159:
157:
154:
138:
118:
110:
94:
86:
70:
63:
59:
50:
46:
40:
33:
19:
5036:Georg Cantor
5031:Paul Bernays
4962:Morse–Kelley
4937:
4870:
4869:Subset
4816:hereditarily
4778:Venn diagram
4736:ordered pair
4651:Intersection
4595:Axiom schema
4417:
4408:
4399:
4390:
4378:
4369:
4360:
4351:
4333:
4324:
4323:Lay (2001),
4315:
4297:
4285:
4276:
4251:
4233:
4215:
4197:
4176:
4164:
4152:
4140:
4128:
4099:
3982:
3979:
3971:group action
3968:
3925:
3910:
3804:
3510:
3473:
3160:
3133:well-defined
2901:
2758:
2375:of integers
1834:
1556:
1502:
1396:
1369:, for every
1357:
1312:. Since its
1273:
1269:
1267:
1262:
1254:
1123:quotient set
1120:
1005:
982:
897:
892:transitivity
552:
466:
384:
381:quotient set
380:
160:
152:
108:
55:
18:Quotient map
5061:Thomas Jech
4904:Alternative
4883:Uncountable
4837:Ultrafilter
4696:Cardinality
4600:replacement
4548:Determinacy
4393:, Wadsworth
4372:, MacMillan
4254:, Freeman,
4183:, pp. 77–78
4181:Maddox 2002
4145:Maddox 2002
4121:Devlin 2004
4104:Devlin 2004
3053:is true if
2684:means that
2245:even number
1314:composition
679:reflexivity
270:is denoted
58:mathematics
5122:Set theory
5101:Categories
5056:Kurt Gödel
5041:Paul Cohen
4878:Transitive
4646:Identities
4630:Complement
4617:Operations
4578:Regularity
4516:Adjunction
4475:Set theory
4363:, PWS-Kent
4190:References
4062:Lie groups
4036:invariants
4032:invariants
3952:linear map
3374:or simply
2898:Invariants
2472:such that
2191:such that
1553:Properties
993:isomorphic
49:Congruence
4989:Paradoxes
4909:Axiomatic
4888:Universal
4864:Singleton
4859:Recursive
4802:Countable
4797:Amorphous
4656:Power set
4573:Power set
4531:dependent
4526:countable
4387:Cupillari
4133:Wolf 1998
3861:∼
3814:∼
3555:∼
3521:∼
3494:→
3458:∼
3436:∼
3414:∼
3386:∼
3358:∼
3301:∼
3283:whenever
3144:∼
3115:∼
3104:invariant
3007:∼
2911:∼
2864:∼
2767:on a set
2669:∼
2495:∼
2439:∼
2333:∼
2228:−
2202:∼
2097:∼
2044:∅
2041:≠
2029:∩
1957:∼
1844:∼
1777:∼
1704:partition
1443:≡
1414:−
1397:congruent
1361:canonical
1278:injection
1270:represent
1261:, or the
1181:↦
959:∼
947:∈
875:∈
837:∼
811:∼
785:∼
752:∈
720:∼
694:∼
662:∈
636:∼
589:∼
557:on a set
504:∼
449:∼
416:∼
354:partition
337:∼
315:∼
190:∼
4993:Problems
4897:Theories
4873:Superset
4849:Infinite
4678:Concepts
4558:Infinity
4482:Overview
4296:(1996),
4171:, p. 127
4135:, p. 178
4042:See also
3913:topology
3901:morphism
3476:function
2164:integers
2063:Examples
1702:forms a
1680:disjoint
1403:divides
1119:(or the
852:for all
769:symmetry
735:for all
709:implies
651:for all
493:topology
5107:Algebra
4931:General
4926:Zermelo
4832:subbase
4814: (
4753:Forcing
4731:Element
4703: (
4681:Methods
4568:Pairing
3740:is the
2892:cliques
2627:of any
1371:integer
1349:section
1167:). The
383:or the
4822:Filter
4812:Finite
4748:Family
4691:Almost
4536:global
4521:Choice
4508:Axioms
4398:Bond,
4340:
4304:
4258:
4240:
4222:
4204:
4079:Setoid
3975:cosets
3778:kernel
3221:; if
2658:, and
2243:is an
2160:modulo
1380:, the
1098:modulo
997:groups
541:, and
4914:Naive
4844:Fuzzy
4807:Empty
4790:types
4741:tuple
4711:Class
4705:large
4666:Union
4583:Union
4420:, MAA
4416:Ash,
4092:Notes
3938:. In
3919:is a
3326:then
1280:from
1222:onto
1202:from
826:then
577:is a
491:or a
107:into
4827:base
4338:ISBN
4302:ISBN
4256:ISBN
4238:ISBN
4220:ISBN
4202:ISBN
3942:, a
3915:, a
3474:Any
2947:and
2890:are
2833:and
2724:are
2704:and
2351:Let
2288:and
2068:Let
1900:and
1880:and
1812:and
1652:and
1399:—if
1391:and
800:and
226:the
131:and
4788:Set
3926:In
3911:In
3879:on
3832:on
3780:of
3744:of
3509:is
3471:".
3131:or
3106:of
2902:If
2759:An
2634:If
2452:on
1706:of
1577:of
1545:by
1541:of
1516:mod
1454:mod
1355:.
1308:to
1147:by
1127:of
985:set
774:if
602:on
553:An
407:by
387:of
356:of
250:in
203:on
62:set
56:In
5103::
4389:,
4111:^
3962:,
3958:,
3930:,
2894:.
2166:,
1549:.
1265:.
894:).
771:),
681:),
545:.
537:,
533:,
529:,
525:,
521:,
4871:·
4855:)
4851:(
4818:)
4707:)
4467:e
4460:t
4453:v
4018:,
4015:X
3995:,
3992:X
3887:Y
3865:Y
3840:X
3818:X
3791:.
3788:f
3764:.
3761:)
3758:x
3755:(
3752:f
3728:]
3725:x
3722:[
3702:,
3699:)
3696:x
3693:(
3690:f
3670:X
3650:x
3630:.
3626:)
3621:2
3617:x
3613:(
3609:f
3606:=
3602:)
3597:1
3593:x
3589:(
3585:f
3563:2
3559:x
3550:1
3546:x
3525:,
3497:Y
3491:X
3488::
3485:f
3389:.
3362:,
3334:f
3314:,
3309:2
3305:x
3296:1
3292:x
3270:)
3265:2
3261:x
3257:(
3253:f
3250:=
3246:)
3241:1
3237:x
3233:(
3229:f
3209:Y
3189:X
3169:f
3147:.
3119:,
3090:P
3070:)
3067:y
3064:(
3061:P
3041:)
3038:x
3035:(
3032:P
3013:,
3010:y
3004:x
2984:X
2964:)
2961:x
2958:(
2955:P
2935:,
2932:X
2870:.
2867:t
2861:s
2841:t
2821:s
2801:,
2798:X
2778:,
2775:X
2736:.
2712:M
2692:L
2672:M
2666:L
2642:X
2631:.
2611:,
2608:b
2604:/
2600:a
2577:)
2574:b
2571:,
2568:a
2565:(
2545:,
2542:c
2539:b
2536:=
2533:d
2530:a
2510:)
2507:d
2504:,
2501:c
2498:(
2492:)
2489:b
2486:,
2483:a
2480:(
2460:X
2418:,
2415:b
2395:)
2392:b
2389:,
2386:a
2383:(
2359:X
2337:.
2328:/
2323:Z
2302:]
2299:1
2296:[
2276:,
2273:]
2270:9
2267:[
2264:,
2261:]
2258:7
2255:[
2231:y
2225:x
2205:y
2199:x
2179:,
2175:Z
2144:.
2141:A
2121:,
2118:A
2076:X
2047:.
2038:]
2035:y
2032:[
2026:]
2023:x
2020:[
1999:]
1996:y
1993:[
1990:=
1987:]
1984:x
1981:[
1960:y
1954:x
1931:,
1928:X
1908:y
1888:x
1868:,
1865:X
1820:y
1800:x
1780:y
1774:x
1754:X
1734:X
1714:X
1690:X
1666:]
1663:y
1660:[
1640:]
1637:x
1634:[
1614:.
1611:]
1608:x
1605:[
1585:X
1565:x
1547:m
1543:a
1525:,
1520:m
1512:a
1488:,
1485:m
1465:.
1461:)
1458:m
1451:(
1446:b
1440:a
1420:;
1417:b
1411:a
1401:m
1393:b
1389:a
1384:m
1378:1
1374:m
1335:,
1332:R
1328:/
1324:X
1310:X
1296:R
1292:/
1288:X
1241:,
1238:R
1234:/
1230:X
1210:X
1190:]
1187:x
1184:[
1178:x
1155:R
1135:X
1107:R
1085:X
1065:,
1062:R
1058:/
1054:X
1034:R
1014:X
968:.
965:}
962:x
956:a
953::
950:X
944:x
941:{
938:=
935:]
932:a
929:[
906:a
890:(
878:X
872:c
869:,
866:b
863:,
860:a
840:c
834:a
814:c
808:b
788:b
782:a
767:(
755:X
749:b
746:,
743:a
723:a
717:b
697:b
691:a
677:(
665:X
659:a
639:a
633:a
610:X
565:X
475:S
453:.
444:/
440:S
420:,
395:S
367:,
364:S
340:.
311:]
307:a
304:[
284:]
281:a
278:[
258:S
238:a
214:,
211:S
169:S
139:b
119:a
95:S
71:S
41:.
34:.
20:)
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