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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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1308:{\displaystyle {\begin{aligned}A_{0}^{*}&=\{(5,0),(6,0),(7,0)\}\\A_{1}^{*}&=\{(5,1),(6,1)\},\\\end{aligned}}}
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labelled (indexed) with the name of the set from which they come. So, an element belonging to both
2760: – Data structure used to hold a value that could take on several different, but fixed, types
2745: – Data structure used to hold a value that could take on several different, but fixed, types
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For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifying
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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\}.}
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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
862:{\textstyle \operatorname {{\bigcup }\!\!\!{\cdot }\,} _{i\in I}A_{i}}
3248:
3070:
2427:
is used for the disjoint union of a family of sets, or the notation
2611:{\displaystyle {\underset {A\in C}{\,\,\bigcup \nolimits ^{*}\!}}A}
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329:
appears twice in the disjoint union, with two different labels.
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36:
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A standard way for building the disjoint union is to define
2641:
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
2768:
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
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2375:{\displaystyle \bigsqcup _{i\in I}A_{i}=A\times I.}
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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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2733: – Equalities for combinations of sets
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95:Learn how and when to remove this message
2480:. It therefore satisfies the associated
2700: – Category-theoretic construction
1810:The elements of the disjoint union are
305:is the set formed from the elements of
2204:In the extreme case where each of the
443:{\textstyle \bigsqcup _{i\in I}A_{i},}
1945:is canonically isomorphic to the set
803:{\textstyle \biguplus _{i\in I}A_{i}}
7:
2731:List of set identities and relations
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
14:
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
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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1:
2789:Graduate Texts in Mathematics
2622:Category theory point of view
2457:of the disjoint union is the
3487:Basic concepts in set theory
2472:, the disjoint union is the
567:, the disjoint union is the
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
3510:
3337: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
989:{\displaystyle A_{i}\to A}
532:(that is, each element of
29:
18:
3138:One-to-one correspondence
2875:
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}
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.
3096: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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3319:Principia Mathematica
3153:Transfinite induction
3012:(i.e. set difference)
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2561:{\displaystyle A_{i}}
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2224:{\displaystyle A_{i}}
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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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3393:Burali-Forti paradox
3148:Set-builder notation
3101:Continuum hypothesis
3041:Symmetric difference
2752:Symmetric difference
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2537:is referred to as a
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3354:Tarski–Grothendieck
2618:is sometimes used.
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2468:In the language of
2446:{\displaystyle A+B}
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2104:
1966:
1674:of sets indexed by
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1486:
1251:
1168:
575:, and thus defined
265:discriminated union
109:
3492:Operations on sets
2943:Limitation of size
2807:Weisstein, Eric W.
2737:Partition of a set
2674:
2651:
2608:
2603:
2558:
2527:
2512:
2498:for more details.
2492:construction. See
2482:universal property
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142:Symbolic statement
52:list of references
3474:
3473:
3383:Russell's paradox
3332:Zermelo–Fraenkel
3233:Dedekind-infinite
3106:Diagonal argument
3005: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}
774:
662:{\displaystyle B}
642:{\displaystyle A}
590:
545:{\displaystyle A}
525:{\displaystyle A}
411:
404:often denoted by
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32:Disjunctive union
3499:
3456:Bertrand Russell
3446:John von Neumann
3431:Abraham Fraenkel
3426:Richard Dedekind
3388:Suslin's problem
3299:Cantor's theorem
3016:De Morgan's laws
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2810:"Disjoint Union"
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573:category of sets
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531:
529:
528:
523:
503:
501:
500:
495:
480:
478:
477:
472:
470:
469:
449:
447:
446:
441:
436:
435:
425:
403:
401:
400:
395:
380:
378:
377:
372:
355:
354:
328:
322:
316:
310:
304:
298:
292:
290:
289:
284:
249:
247:
246:
241:
239:
235:
234:
233:
194:
176:
175:
165:
117:
110:
100:
93:
89:
86:
80:
75:this article by
66:inline citations
45:
44:
37:
3509:
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3501:
3500:
3498:
3497:
3496:
3477:
3476:
3475:
3470:
3397:
3376:
3360:
3325:New Foundations
3272:
3162:
3081:Cardinal number
3064:
3050:
2991:
2882:
2873:
2857:
2852:
2805:
2804:
2799:
2779:
2776:
2767:
2746:
2713:
2694:
2663:
2662:
2643:
2642:
2628:category theory
2624:
2592:
2580:
2577:
2570:
2569:
2548:
2543:
2542:
2507:
2506:
2470:category theory
2429:
2428:
2407:
2386:
2385:
2347:
2326:
2325:
2306:
2305:
2286:
2285:
2253:
2252:
2233:
2232:
2211:
2206:
2205:
2181:
2176:
2175:
2154:
2149:
2148:
2117:
2116:
2085:
2084:
2056:
2055:
2034:
2029:
2028:
1999:
1974:
1970:
1947:
1946:
1925:
1920:
1919:
1897:
1896:
1875:
1870:
1869:
1850:
1849:
1815:
1814:
1782:
1757:
1753:
1724:
1703:
1702:
1676:
1675:
1633:
1632:
1628:
1623:
1622:
1619:
1459:
1446:
1441:
1440:
1419:
1406:
1401:
1400:
1376:
1371:
1370:
1339:
1338:
1319:
1318:
1302:
1301:
1252:
1234:
1233:
1169:
1145:
1144:
1102:
1097:
1096:
1051:
1046:
1045:
1042:
998:
997:
970:
965:
964:
940:
929:
928:
897:
896:
874:
873:
849:
817:
812:
811:
790:
769:
768:
727:
726:
701:
700:
675:
674:
651:
650:
631:
630:
626:is often used.
606:
585:
584:
565:category theory
534:
533:
514:
513:
483:
482:
461:
456:
455:
427:
406:
405:
383:
382:
346:
338:
337:
324:
318:
312:
306:
300:
294:
269:
268:
225:
200:
196:
167:
146:
145:
101:
90:
84:
81:
70:
56:related reading
46:
42:
35:
28:
17:
12:
11:
5:
3507:
3506:
3503:
3495:
3494:
3489:
3479:
3478:
3472:
3471:
3469:
3468:
3463:
3461:Thoralf Skolem
3458:
3453:
3448:
3443:
3438:
3433:
3428:
3423:
3418:
3413:
3407:
3405:
3399:
3398:
3396:
3395:
3390:
3385:
3379:
3377:
3375:
3374:
3371:
3365:
3362:
3361:
3359:
3358:
3357:
3356:
3351:
3346:
3345:
3344:
3329:
3328:
3327:
3315:
3314:
3313:
3302:
3301:
3296:
3291:
3286:
3280:
3278:
3274:
3273:
3271:
3270:
3265:
3260:
3255:
3246:
3241:
3236:
3226:
3221:
3220:
3219:
3214:
3209:
3199:
3189:
3184:
3179:
3173:
3171:
3164:
3163:
3161:
3160:
3155:
3150:
3145:
3143:Ordinal number
3140:
3135:
3130:
3125:
3124:
3123:
3118:
3108:
3103:
3098:
3093:
3088:
3078:
3073:
3067:
3065:
3063:
3062:
3059:
3055:
3052:
3051:
3049:
3048:
3043:
3038:
3033:
3028:
3023:
3021:Disjoint union
3018:
3013:
3007:
3001:
2999:
2993:
2992:
2990:
2989:
2988:
2987:
2982:
2971:
2970:
2968:Martin's axiom
2965:
2960:
2955:
2950:
2945:
2940:
2935:
2933:Extensionality
2930:
2925:
2920:
2919:
2918:
2913:
2908:
2898:
2892:
2890:
2884:
2883:
2876:
2874:
2872:
2871:
2865:
2863:
2859:
2858:
2853:
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:
13:
10:
9:
6:
4:
3:
2:
3505:
3504:
3493:
3490:
3488:
3485:
3484:
3482:
3467:
3466:Ernst Zermelo
3464:
3462:
3459:
3457:
3454:
3452:
3451:Willard Quine
3449:
3447:
3444:
3442:
3439:
3437:
3434:
3432:
3429:
3427:
3424:
3422:
3419:
3417:
3414:
3412:
3409:
3408:
3406:
3404:
3403:Set theorists
3400:
3394:
3391:
3389:
3386:
3384:
3381:
3380:
3378:
3372:
3370:
3367:
3366:
3363:
3355:
3352:
3350:
3349:Kripke–Platek
3347:
3343:
3340:
3339:
3338:
3335:
3334:
3333:
3330:
3326:
3323:
3322:
3321:
3320:
3316:
3312:
3309:
3308:
3307:
3304:
3303:
3300:
3297:
3295:
3292:
3290:
3287:
3285:
3282:
3281:
3279:
3275:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3252:
3247:
3245:
3242:
3240:
3237:
3234:
3230:
3227:
3225:
3222:
3218:
3215:
3213:
3210:
3208:
3205:
3204:
3203:
3200:
3197:
3193:
3190:
3188:
3185:
3183:
3180:
3178:
3175:
3174:
3172:
3169:
3165:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3126:
3122:
3119:
3117:
3114:
3113:
3112:
3109:
3107:
3104:
3102:
3099:
3097:
3094:
3092:
3089:
3086:
3082:
3079:
3077:
3074:
3072:
3069:
3068:
3066:
3060:
3057:
3056:
3053:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3027:
3024:
3022:
3019:
3017:
3014:
3011:
3008:
3006:
3003:
3002:
3000:
2998:
2994:
2986:
2985:specification
2983:
2981:
2978:
2977:
2976:
2973:
2972:
2969:
2966:
2964:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
2944:
2941:
2939:
2936:
2934:
2931:
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:
2773:
2765:
2762:
2759:
2756:
2753:
2750:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2711:
2708:
2705:
2702:
2699:
2696:
2695:
2691:
2689:
2687:
2671:
2668:
2648:
2639:
2635:
2633:
2629:
2621:
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:
1616:
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:
1039:
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:
3416:Georg Cantor
3411:Paul Bernays
3342:Morse–Kelley
3317:
3250:
3249:Subset
3196:hereditarily
3158:Venn diagram
3116:ordered pair
3031:Intersection
3020:
2975: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
3441:Thomas Jech
3284:Alternative
3263:Uncountable
3217:Ultrafilter
3076:Cardinality
2980:replacement
2928:Determinacy
2781:Lang, Serge
2455:cardinality
1915:came from.
257:mathematics
77:introducing
3481:Categories
3436:Kurt Gödel
3421:Paul Cohen
3258:Transitive
3026:Identities
3010:Complement
2997:Operations
2958:Regularity
2896:Adjunction
2855:Set theory
2774:References
2684:to denote
2201:are not.
927:such that
136:Set theory
3369:Paradoxes
3289:Axiomatic
3268:Universal
3244:Singleton
3239:Recursive
3182:Countable
3177:Amorphous
3036:Power set
2953: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:⨆
3373:Problems
3277:Theories
3253:Superset
3229:Infinite
3058:Concepts
2938:Infinity
2862:Overview
2783:(2004),
2743:Sum type
2692:See also
454:of each
450:with an
336:of sets
3311:General
3306:Zermelo
3212:subbase
3194: (
3133:Forcing
3111:Element
3083: (
3061:Methods
2948:Pairing
2785:Algebra
2488:of the
2476:in the
1040:Example
571:of the
73:improve
3202:Filter
3192:Finite
3128:Family
3071:Almost
2916:global
2901:Choice
2888:Axioms
2795:
1670:be an
506:images
259:, the
3294:Naive
3224:Fuzzy
3187:Empty
3170:types
3121:tuple
3091:Class
3085:large
3046:Union
2963:Union
1848:Here
577:up to
558:union
481:into
132:Field
58:, or
3207: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
3168:Set
2626:In
2541:of
2459:sum
2284:of
1337:in
996:as
810:or
725:or
673:as
563:In
512:of
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