43:
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As such, the disjoint union is defined up to an isomorphism, and the above definition is just one realization of the coproduct, among others. When the sets are pairwise disjoint, the usual union is another realization of the coproduct. This justifies the second definition in the lead.
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2712: – space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology
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1308:{\displaystyle {\begin{aligned}A_{0}^{*}&=\{(5,0),(6,0),(7,0)\}\\A_{1}^{*}&=\{(5,1),(6,1)\},\\\end{aligned}}}
94:
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labelled (indexed) with the name of the set from which they come. So, an element belonging to both
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For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifying
2453:
for the disjoint union of two sets. This notation is meant to be suggestive of the fact that the
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where the second element in each pair matches the subscript of the origin set (for example, the
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It is possible to index the set elements according to set origin by forming the associated sets
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This article is about the operation on sets. For the computer science meaning of the term, see
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1607:{\displaystyle A_{0}\sqcup A_{1}=A_{0}^{*}\cup A_{1}^{*}=\{(5,0),(6,0),(7,0),(5,1),(6,1)\}.}
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1803:{\displaystyle \bigsqcup _{i\in I}A_{i}=\bigcup _{i\in I}\left\{(x,i):x\in A_{i}\right\}.}
8:
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243:{\displaystyle \bigsqcup _{i\in I}A_{i}=\bigcup _{i\in I}\left\{(x,i):x\in A_{i}\right\}}
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of the cardinalities of the terms in the family. Compare this to the notation for the
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belongs to exactly one of these images). A disjoint union of a family of
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862:{\textstyle \operatorname {{\bigcup }\!\!\!{\cdot }\,} _{i\in I}A_{i}}
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is used for the disjoint union of a family of sets, or the notation
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2611:{\displaystyle {\underset {A\in C}{\,\,\bigcup \nolimits ^{*}\!}}A}
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appears twice in the disjoint union, with two different labels.
3255:
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A standard way for building the disjoint union is to define
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This categorical aspect of the disjoint union explains why
2020:{\displaystyle A_{i}^{*}=\left\{(x,i):x\in A_{i}\right\}.}
760:{\displaystyle A\operatorname {{\cup }\!\!\!{\cdot }\,} B}
2721: – Combining the vertex and edge sets of two graphs
2804:
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Pages displaying short descriptions of redirect targets
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Pages displaying short descriptions of redirect targets
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2739: – Mathematical ways to group elements of a set
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Pages displaying wikidata descriptions as a fallback
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is canonically embedded in the disjoint union. For
2375:{\displaystyle \bigsqcup _{i\in I}A_{i}=A\times I.}
2727: – Set of elements common to all of some sets
2706: – Special case of colimit in category theory
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2484:. This also means that the disjoint union is the
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2027:Through this isomorphism, one may consider that
64:but its sources remain unclear because it lacks
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699:. Some authors use the alternative notation
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2754: – Elements in exactly one of two sets
2733: – Equalities for combinations of sets
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1663:{\displaystyle \left(A_{i}:i\in I\right)}
832:
749:
95:Learn how and when to remove this message
2480:. It therefore satisfies the associated
1616:
2700: – Category-theoretic construction
1810:The elements of the disjoint union are
305:is the set formed from the elements of
3486:
2204:In the extreme case where each of the
443:{\textstyle \bigsqcup _{i\in I}A_{i},}
2827:
2805:
1945:is canonically isomorphic to the set
803:{\textstyle \biguplus _{i\in I}A_{i}}
2779:
2731:List of set identities and relations
36:
2630:the disjoint union is defined as a
2581:
2420:{\displaystyle \sum _{i\in I}A_{i}}
1439:can then be calculated as follows:
619:{\textstyle \coprod _{i\in I}A_{i}}
23:. For the operation on graphs, see
13:
14:
3510:
1432:{\displaystyle A_{0}\sqcup A_{1}}
583:. In this context, the notation
16:In mathematics, operation on sets
2877:
41:
2661:is frequently used, instead of
1088:{\displaystyle A_{0}=\{5,6,7\}}
1030:{\displaystyle x\mapsto (x,i).}
629:The disjoint union of two sets
2853:
2147:are disjoint even if the sets
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1136:{\displaystyle A_{1}=\{5,6\}.}
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767:(along with the corresponding
374:{\displaystyle (A_{i}:i\in I)}
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2789:Graduate Texts in Mathematics
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2622:Category theory point of view
2457:of the disjoint union is the
3494:Basic concepts in set theory
2472:, the disjoint union is the
567:, the disjoint union is the
7:
2691:
2677:{\displaystyle \bigsqcup ,}
2384:Occasionally, the notation
2231:is equal to some fixed set
956:{\displaystyle x\in A_{i},}
508:of these injections form a
10:
3515:
3344:von Neumann–Bernays–Gödel
2634:in the category of sets.
2280:the disjoint union is the
1701:of this family is the set
1399:etc.). The disjoint union
1039:
989:{\displaystyle A_{i}\to A}
532:(that is, each element of
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18:
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3145:One-to-one correspondence
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2861:
2725:Intersection (set theory)
2710:Disjoint union (topology)
2530:{\displaystyle A_{i}^{*}}
2140:{\displaystyle A_{j}^{*}}
2108:{\displaystyle A_{i}^{*}}
1369:matches the subscript in
718:{\displaystyle A\uplus B}
692:{\displaystyle A\sqcup B}
286:{\displaystyle A\sqcup B}
141:
131:
121:
112:
2764:Union (computer science)
2719:Disjoint union of graphs
2654:{\displaystyle \coprod }
2076:{\displaystyle i\neq j,}
50:This article includes a
30:Not to be confused with
25:disjoint union of graphs
2273:{\displaystyle i\in I,}
332:A disjoint union of an
79:more precise citations.
3103:Constructible universe
2923:Constructibility (V=L)
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2465:of a family of sets.
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1841:{\displaystyle (x,i).}
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1392:{\displaystyle A_{0},}
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554:pairwise disjoint sets
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3326:Principia Mathematica
3160:Transfinite induction
3019:(i.e. set difference)
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1938:{\displaystyle A_{i}}
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1617:Set theory definition
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474:{\displaystyle A_{i}}
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3400:Burali-Forti paradox
3155:Set-builder notation
3108:Continuum hypothesis
3048:Symmetric difference
2752:Symmetric difference
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2468:In the language of
2446:{\displaystyle A+B}
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1966:
1674:of sets indexed by
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1251:
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575:, and thus defined
265:discriminated union
109:
3499:Operations on sets
2950:Limitation of size
2807:Weisstein, Eric W.
2737:Partition of a set
2674:
2651:
2608:
2603:
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2498:for more details.
2492:construction. See
2482:universal property
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142:Symbolic statement
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52:list of references
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3390:Russell's paradox
3339:Zermelo–Fraenkel
3240:Dedekind-infinite
3113:Diagonal argument
3012:Cartesian product
2869:Set (mathematics)
2798:978-0-387-95385-4
2576:
2568:and the notation
2503:abuse of notation
2490:Cartesian product
2463:Cartesian product
2391:
2331:
2317:{\displaystyle I}
2297:{\displaystyle A}
2282:Cartesian product
2244:{\displaystyle A}
1918:Each of the sets
1908:{\displaystyle x}
1861:{\displaystyle i}
1737:
1708:
1330:{\displaystyle 0}
885:{\displaystyle A}
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662:{\displaystyle B}
642:{\displaystyle A}
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545:{\displaystyle A}
525:{\displaystyle A}
411:
404:often denoted by
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32:Disjunctive union
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3463:Bertrand Russell
3453:John von Neumann
3438:Abraham Fraenkel
3433:Richard Dedekind
3395:Suslin's problem
3306:Cantor's theorem
3023:De Morgan's laws
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573:category of sets
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75:this article by
66:inline citations
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2573:
2570:
2569:
2552:
2548:
2546:
2543:
2542:
2521:
2516:
2510:
2507:
2506:
2470:category theory
2432:
2429:
2428:
2411:
2407:
2395:
2389:
2386:
2385:
2351:
2347:
2335:
2329:
2326:
2325:
2309:
2306:
2305:
2289:
2286:
2285:
2256:
2253:
2252:
2236:
2233:
2232:
2215:
2211:
2209:
2206:
2205:
2185:
2181:
2179:
2176:
2175:
2158:
2154:
2152:
2149:
2148:
2131:
2126:
2120:
2117:
2116:
2099:
2094:
2088:
2085:
2084:
2059:
2056:
2055:
2038:
2034:
2032:
2029:
2028:
2003:
1999:
1974:
1970:
1961:
1956:
1950:
1947:
1946:
1929:
1925:
1923:
1920:
1919:
1900:
1897:
1896:
1879:
1875:
1873:
1870:
1869:
1853:
1850:
1849:
1818:
1815:
1814:
1786:
1782:
1757:
1753:
1741:
1728:
1724:
1712:
1706:
1703:
1702:
1679:
1676:
1675:
1637:
1633:
1632:
1628:
1626:
1623:
1622:
1619:
1499:
1494:
1481:
1476:
1463:
1459:
1450:
1446:
1444:
1441:
1440:
1423:
1419:
1410:
1406:
1404:
1401:
1400:
1380:
1376:
1374:
1371:
1370:
1342:
1339:
1338:
1322:
1319:
1318:
1302:
1301:
1252:
1246:
1241:
1234:
1233:
1169:
1163:
1158:
1150:
1148:
1145:
1144:
1106:
1102:
1100:
1097:
1096:
1055:
1051:
1049:
1046:
1045:
1042:
1001:
998:
997:
974:
970:
968:
965:
964:
944:
940:
932:
929:
928:
900:
897:
896:
877:
874:
873:
853:
849:
834:
827:
819:
818:
817:
815:
812:
811:
794:
790:
778:
772:
769:
768:
744:
736:
735:
730:
727:
726:
704:
701:
700:
678:
675:
674:
654:
651:
650:
634:
631:
630:
626:is often used.
610:
606:
594:
588:
585:
584:
565:category theory
537:
534:
533:
517:
514:
513:
486:
483:
482:
465:
461:
459:
456:
455:
431:
427:
415:
409:
406:
405:
386:
383:
382:
350:
346:
341:
338:
337:
324:
318:
312:
306:
300:
294:
272:
269:
268:
229:
225:
200:
196:
184:
171:
167:
155:
149:
146:
145:
101:
90:
84:
81:
70:
56:related reading
46:
42:
35:
28:
17:
12:
11:
5:
3512:
3502:
3501:
3496:
3479:
3478:
3476:
3475:
3470:
3468:Thoralf Skolem
3465:
3460:
3455:
3450:
3445:
3440:
3435:
3430:
3425:
3420:
3414:
3412:
3406:
3405:
3403:
3402:
3397:
3392:
3386:
3384:
3382:
3381:
3378:
3372:
3369:
3368:
3366:
3365:
3364:
3363:
3358:
3353:
3352:
3351:
3336:
3335:
3334:
3322:
3321:
3320:
3309:
3308:
3303:
3298:
3293:
3287:
3285:
3281:
3280:
3278:
3277:
3272:
3267:
3262:
3253:
3248:
3243:
3233:
3228:
3227:
3226:
3221:
3216:
3206:
3196:
3191:
3186:
3180:
3178:
3171:
3170:
3168:
3167:
3162:
3157:
3152:
3150:Ordinal number
3147:
3142:
3137:
3132:
3131:
3130:
3125:
3115:
3110:
3105:
3100:
3095:
3085:
3080:
3074:
3072:
3070:
3069:
3066:
3062:
3059:
3058:
3056:
3055:
3050:
3045:
3040:
3035:
3030:
3028:Disjoint union
3025:
3020:
3014:
3008:
3006:
3000:
2999:
2997:
2996:
2995:
2994:
2989:
2978:
2977:
2975:Martin's axiom
2972:
2967:
2962:
2957:
2952:
2947:
2942:
2940:Extensionality
2937:
2936:
2935:
2925:
2920:
2919:
2918:
2913:
2908:
2898:
2892:
2890:
2884:
2883:
2876:
2874:
2872:
2871:
2865:
2863:
2859:
2858:
2851:
2850:
2843:
2836:
2828:
2822:
2821:
2802:
2797:
2775:
2772:
2771:
2770:
2761:
2755:
2749:
2740:
2734:
2728:
2722:
2716:
2707:
2701:
2693:
2690:
2673:
2670:
2650:
2623:
2620:
2607:
2601:
2598:
2595:
2587:
2583:
2555:
2551:
2540:
2524:
2519:
2515:
2442:
2439:
2436:
2414:
2410:
2404:
2401:
2398:
2394:
2371:
2368:
2365:
2362:
2359:
2354:
2350:
2344:
2341:
2338:
2334:
2313:
2293:
2269:
2266:
2263:
2260:
2240:
2218:
2214:
2188:
2184:
2161:
2157:
2134:
2129:
2125:
2102:
2097:
2093:
2072:
2069:
2066:
2063:
2041:
2037:
2016:
2012:
2006:
2002:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1973:
1969:
1964:
1959:
1955:
1932:
1928:
1904:
1882:
1878:
1857:
1837:
1834:
1831:
1828:
1825:
1822:
1799:
1795:
1789:
1785:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1756:
1750:
1747:
1744:
1740:
1736:
1731:
1727:
1721:
1718:
1715:
1711:
1699:disjoint union
1686:
1683:
1672:indexed family
1658:
1654:
1651:
1648:
1645:
1640:
1636:
1631:
1621:Formally, let
1618:
1615:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1502:
1497:
1493:
1489:
1484:
1479:
1475:
1471:
1466:
1462:
1458:
1453:
1449:
1426:
1422:
1418:
1413:
1409:
1388:
1383:
1379:
1358:
1355:
1352:
1349:
1346:
1326:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1253:
1249:
1244:
1240:
1236:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1170:
1166:
1161:
1157:
1153:
1152:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1109:
1105:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1058:
1054:
1041:
1038:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
985:
982:
977:
973:
952:
947:
943:
939:
936:
916:
913:
910:
907:
904:
892:as the set of
881:
856:
852:
848:
843:
840:
837:
830:
822:
797:
793:
787:
784:
781:
777:
756:
753:
747:
739:
734:
714:
711:
708:
688:
685:
682:
671:infix notation
658:
638:
613:
609:
603:
600:
597:
593:
541:
521:
504:such that the
493:
490:
468:
464:
439:
434:
430:
424:
421:
418:
414:
393:
390:
370:
367:
364:
361:
358:
353:
349:
345:
334:indexed family
282:
279:
276:
261:disjoint union
251:
250:
238:
232:
228:
224:
221:
218:
215:
212:
209:
206:
203:
199:
193:
190:
187:
183:
179:
174:
170:
164:
161:
158:
154:
143:
139:
138:
133:
129:
128:
123:
119:
118:
108:Disjoint union
103:
102:
60:external links
49:
47:
40:
15:
9:
6:
4:
3:
2:
3511:
3500:
3497:
3495:
3492:
3491:
3489:
3474:
3473:Ernst Zermelo
3471:
3469:
3466:
3464:
3461:
3459:
3458:Willard Quine
3456:
3454:
3451:
3449:
3446:
3444:
3441:
3439:
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3419:
3416:
3415:
3413:
3411:
3410:Set theorists
3407:
3401:
3398:
3396:
3393:
3391:
3388:
3387:
3385:
3379:
3377:
3374:
3373:
3370:
3362:
3359:
3357:
3356:Kripke–Platek
3354:
3350:
3347:
3346:
3345:
3342:
3341:
3340:
3337:
3333:
3330:
3329:
3328:
3327:
3323:
3319:
3316:
3315:
3314:
3311:
3310:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3288:
3286:
3282:
3276:
3273:
3271:
3268:
3266:
3263:
3261:
3259:
3254:
3252:
3249:
3247:
3244:
3241:
3237:
3234:
3232:
3229:
3225:
3222:
3220:
3217:
3215:
3212:
3211:
3210:
3207:
3204:
3200:
3197:
3195:
3192:
3190:
3187:
3185:
3182:
3181:
3179:
3176:
3172:
3166:
3163:
3161:
3158:
3156:
3153:
3151:
3148:
3146:
3143:
3141:
3138:
3136:
3133:
3129:
3126:
3124:
3121:
3120:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3093:
3089:
3086:
3084:
3081:
3079:
3076:
3075:
3073:
3067:
3064:
3063:
3060:
3054:
3051:
3049:
3046:
3044:
3041:
3039:
3036:
3034:
3031:
3029:
3026:
3024:
3021:
3018:
3015:
3013:
3010:
3009:
3007:
3005:
3001:
2993:
2992:specification
2990:
2988:
2985:
2984:
2983:
2980:
2979:
2976:
2973:
2971:
2968:
2966:
2963:
2961:
2958:
2956:
2953:
2951:
2948:
2946:
2943:
2941:
2938:
2934:
2931:
2930:
2929:
2926:
2924:
2921:
2917:
2914:
2912:
2909:
2907:
2904:
2903:
2902:
2899:
2897:
2894:
2893:
2891:
2889:
2885:
2880:
2870:
2867:
2866:
2864:
2860:
2856:
2849:
2844:
2842:
2837:
2835:
2830:
2829:
2826:
2817:
2816:
2811:
2808:
2803:
2800:
2794:
2790:
2786:
2782:
2778:
2777:
2765:
2762:
2759:
2756:
2753:
2750:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2711:
2708:
2705:
2702:
2699:
2696:
2695:
2689:
2687:
2671:
2668:
2648:
2639:
2635:
2633:
2629:
2619:
2605:
2599:
2596:
2593:
2585:
2553:
2549:
2538:
2522:
2517:
2513:
2504:
2499:
2497:
2496:
2491:
2487:
2483:
2479:
2475:
2471:
2466:
2464:
2460:
2456:
2440:
2437:
2434:
2412:
2408:
2402:
2399:
2396:
2392:
2382:
2369:
2366:
2363:
2360:
2357:
2352:
2348:
2342:
2339:
2336:
2332:
2311:
2291:
2283:
2267:
2264:
2261:
2258:
2238:
2216:
2212:
2202:
2186:
2182:
2159:
2155:
2132:
2127:
2123:
2100:
2095:
2091:
2070:
2067:
2064:
2061:
2039:
2035:
2014:
2010:
2004:
2000:
1996:
1993:
1990:
1984:
1981:
1978:
1971:
1967:
1962:
1957:
1953:
1930:
1926:
1916:
1902:
1880:
1876:
1855:
1835:
1829:
1826:
1823:
1813:
1812:ordered pairs
1797:
1793:
1787:
1783:
1779:
1776:
1773:
1767:
1764:
1761:
1754:
1748:
1745:
1742:
1738:
1734:
1729:
1725:
1719:
1716:
1713:
1709:
1700:
1684:
1681:
1673:
1656:
1652:
1649:
1646:
1643:
1638:
1634:
1629:
1614:
1601:
1592:
1589:
1586:
1580:
1574:
1571:
1568:
1562:
1556:
1553:
1550:
1544:
1538:
1535:
1532:
1526:
1520:
1517:
1514:
1505:
1500:
1495:
1491:
1487:
1482:
1477:
1473:
1469:
1464:
1460:
1456:
1451:
1447:
1424:
1420:
1416:
1411:
1407:
1386:
1381:
1377:
1353:
1350:
1347:
1324:
1315:
1298:
1289:
1286:
1283:
1277:
1271:
1268:
1265:
1256:
1254:
1247:
1242:
1238:
1224:
1221:
1218:
1212:
1206:
1203:
1200:
1194:
1188:
1185:
1182:
1173:
1171:
1164:
1159:
1155:
1130:
1124:
1121:
1118:
1112:
1107:
1103:
1079:
1076:
1073:
1070:
1067:
1061:
1056:
1052:
1037:
1024:
1018:
1015:
1012:
1003:
983:
975:
971:
950:
945:
941:
937:
934:
911:
908:
905:
895:
894:ordered pairs
879:
870:
854:
850:
846:
841:
838:
835:
828:
820:
795:
791:
785:
782:
779:
775:
754:
751:
745:
737:
732:
712:
709:
706:
686:
683:
680:
672:
656:
636:
627:
611:
607:
601:
598:
595:
591:
582:
578:
574:
570:
566:
561:
559:
555:
539:
519:
511:
507:
491:
488:
466:
462:
453:
437:
432:
428:
422:
419:
416:
412:
391:
388:
365:
362:
359:
356:
351:
347:
335:
330:
327:
321:
315:
309:
303:
297:
280:
277:
274:
266:
262:
258:
236:
230:
226:
222:
219:
216:
210:
207:
204:
197:
191:
188:
185:
181:
177:
172:
168:
162:
159:
156:
152:
144:
140:
137:
134:
130:
127:
126:Set operation
124:
120:
116:
111:
99:
96:
88:
78:
74:
68:
67:
61:
57:
53:
48:
39:
38:
33:
26:
22:
3423:Georg Cantor
3418:Paul Bernays
3349:Morse–Kelley
3324:
3257:
3256:Subset
3203:hereditarily
3165:Venn diagram
3123:ordered pair
3038:Intersection
3027:
2982:Axiom schema
2813:
2784:
2758:Tagged union
2704:Direct limit
2685:
2640:
2636:
2625:
2500:
2493:
2467:
2383:
2203:
1917:
1895:the element
1698:
1620:
1316:
1043:
871:
628:
562:
331:
325:
319:
313:
307:
301:
295:
293:of the sets
264:
260:
254:
91:
85:January 2022
82:
71:Please help
63:
21:Tagged union
3448:Thomas Jech
3291:Alternative
3270:Uncountable
3224:Ultrafilter
3083:Cardinality
2987:replacement
2928:Determinacy
2781:Lang, Serge
2455:cardinality
1915:came from.
257:mathematics
77:introducing
3488:Categories
3443:Kurt Gödel
3428:Paul Cohen
3265:Transitive
3033:Identities
3017:Complement
3004:Operations
2965:Regularity
2933:projective
2896:Adjunction
2855:Set theory
2774:References
2684:to denote
2201:are not.
927:such that
136:Set theory
3376:Paradoxes
3296:Axiomatic
3275:Universal
3251:Singleton
3246:Recursive
3189:Countable
3184:Amorphous
3043:Power set
2960:Power set
2911:dependent
2906:countable
2815:MathWorld
2698:Coproduct
2686:coproduct
2669:⨆
2649:∐
2632:coproduct
2597:∈
2586:∗
2582:⋃
2523:∗
2495:Coproduct
2474:coproduct
2400:∈
2393:∑
2364:×
2340:∈
2333:⨆
2262:∈
2251:for each
2133:∗
2101:∗
2083:the sets
2065:≠
1997:∈
1963:∗
1780:∈
1746:∈
1739:⋃
1717:∈
1710:⨆
1650:∈
1501:∗
1488:∪
1483:∗
1457:⊔
1417:⊔
1248:∗
1165:∗
1007:↦
981:→
938:∈
847:
839:∈
829:⋅
821:⋃
783:∈
776:⨄
752:
746:⋅
738:∪
710:⊎
684:⊔
599:∈
592:∐
581:bijection
569:coproduct
556:is their
510:partition
452:injection
420:∈
413:⨆
381:is a set
363:∈
278:⊔
223:∈
189:∈
182:⋃
160:∈
153:⨆
3380:Problems
3284:Theories
3260:Superset
3236:Infinite
3065:Concepts
2945:Infinity
2862:Overview
2783:(2004),
2743:Sum type
2692:See also
454:of each
450:with an
336:of sets
3318:General
3313:Zermelo
3219:subbase
3201: (
3140:Forcing
3118:Element
3090: (
3068:Methods
2955:Pairing
2785:Algebra
2488:of the
2476:in the
1040:Example
571:of the
73:improve
3209:Filter
3199:Finite
3135:Family
3078:Almost
2916:global
2901:Choice
2888:Axioms
2795:
1670:be an
506:images
259:, the
3301:Naive
3231:Fuzzy
3194:Empty
3177:types
3128:tuple
3098:Class
3092:large
3053:Union
2970:Union
1848:Here
577:up to
558:union
481:into
132:Field
58:, or
3214:base
2793:ISBN
2539:copy
2304:and
2174:and
2115:and
1697:The
1095:and
869:).
649:and
323:and
311:and
299:and
263:(or
122:Type
3175:Set
2626:In
2541:of
2459:sum
2284:of
1337:in
996:as
810:or
725:or
673:as
563:In
512:of
255:In
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