202:
210:
1098:
206:
1138:
104:
33:
96:
218:
92:
41:
190:
166:
135:
233:
214:
116:
112:
21:
127:
25:
1111:
174:
895:
186:
120:
222:
194:
182:
159:
155:
139:
226:
986:
620:
198:
37:
29:
1132:
48:
143:
100:
169:
courses sometimes leave students with the impression that the subject matter of
178:
170:
173:
is no more than these properties. For more about elementary set theory, see
68:
56:
59:, denoted {}. The algebra of sets describes the properties of all possible
95:
under union, intersection, and set complement. The algebra of sets is an
60:
47:
These properties assume the existence of at least two sets: a given
1114:(1904) "Sets of independent postulates for the algebra of logic,"
131:
236:, denoted by the infix operator "\". The "relative complement of
189:. For an introduction to set theory at a higher level, see also
130:, but can be derived from a small number of properties taken as
232:
The properties below include a defined binary operation,
134:. A "*" follows the algebra of sets interpretation of
1116:Transactions of the American Mathematical Society
158:. The properties followed by "L" interpret the
20:are some of the elementary properties of the
8:
107:, with union, intersection, set complement,
1101: – Equalities for combinations of sets
142:. These properties can be visualized with
126:The properties below are stated without
1125:. Addison-Wesley. Dover reprint, 1999.
146:. They also follow from the fact that
18:simple theorems in the algebra of sets
7:
1123:Boolean Algebra and Its Applications
1099:List of set identities and relations
211:Cantor's first uncountability proof
203:Cantor–Bernstein–Schroeder theorem
14:
138:(1904) classic postulate set for
111:, and {} interpreting Boolean
1:
1155:
207:Cantor's diagonal argument
123:, 1, and 0, respectively.
36:(infix operator: ∩), and
1121:Whitesitt, J. E. (1961)
973:. Some properties of ⊆:
342: ∩ {} = {};
219:well-ordering theorem
191:axiomatic set theory
167:discrete mathematics
234:relative complement
1139:Operations on sets
348: ∪ {} =
325: \ {} =
1112:Edward Huntington
896:distributive laws
252:, is defined as (
1146:
1044:
1037:
669:
664:
657:
651:
642:
458:
454:
435:
404:
387:
317:
304:
272:
266:
261:
215:Cantor's theorem
187:naive set theory
1154:
1153:
1149:
1148:
1147:
1145:
1144:
1143:
1129:
1128:
1108:
1095:
1056:if and only if
1042:
1035:
1032:if and only if
1010:if and only if
969:
667:
662:
655:
649:
640:
499:. For any sets
495:
456:
452:
433:
402:
385:
315:
302:
289:and any subset
281:
270:
264:
259:
223:axiom of choice
195:cardinal number
183:algebra of sets
156:Boolean lattice
140:Boolean algebra
105:Boolean algebra
12:
11:
5:
1152:
1150:
1142:
1141:
1131:
1130:
1127:
1126:
1119:
1107:
1104:
1103:
1102:
1094:
1091:
1090:
1089:
1088:
1087:
1065:
1047:
1023:
1001:
987:if and only if
967:
966:
965:
964:
933:
892:
891:
890:
889:
858:
819:
789:
759:
729:
702:
675:
635:
621:if and only if
610:
587:
569:
551:
533:
493:
492:
478:
464:
448:
438:
421:
411:
398:
381:
367:
353:
343:
337:
330:
320:
310:
199:ordinal number
97:interpretation
38:set complement
30:infix operator
13:
10:
9:
6:
4:
3:
2:
1151:
1140:
1137:
1136:
1134:
1124:
1120:
1117:
1113:
1110:
1109:
1105:
1100:
1097:
1096:
1092:
1085:
1081:
1077:
1073:
1069:
1066:
1063:
1059:
1055:
1051:
1048:
1045:
1038:
1031:
1027:
1024:
1021:
1017:
1013:
1009:
1005:
1002:
999:
995:
991:
988:
985:
981:
978:
977:
976:
975:
974:
972:
971:PROPOSITION 3
962:
958:
954:
950:
946:
942:
938:
934:
931:
927:
923:
919:
915:
911:
907:
903:
902:
901:
900:
899:
897:
887:
884: \
883:
879:
876: ∪
875:
871:
867:
864: \
863:
859:
856:
853: \
852:
848:
844:
840:
837: ∩
836:
832:
828:
825: \
824:
820:
817:
814: ∩
813:
809:
806: \
805:
801:
798: \
797:
793:
790:
787:
784: \
783:
779:
776: \
775:
771:
768: ∪
767:
763:
760:
757:
754: \
753:
749:
746: \
745:
741:
738: ∩
737:
733:
730:
727:
724: ∪
723:
719:
715:
711:
708: ∪
707:
703:
700:
697: ∩
696:
692:
688:
684:
681: ∩
680:
676:
673:
665:
658:
647:
643:
636:
633:
629:
625:
622:
618:
614:
611:
608:
604:
600:
596:
592:
588:
585:
581:
577:
573:
570:
567:
563:
559:
555:
552:
549:
545:
541:
537:
534:
531:
527:
523:
519:
516:
515:
514:
513:
512:
510:
506:
502:
498:
497:PROPOSITION 2
490:
486:
482:
479:
476:
472:
468:
465:
462:
455:
449:
446:
442:
439:
436:
429:
425:
422:
419:
415:
412:
409:
405:
399:
396:
392:
388:
382:
379:
375:
371:
368:
365:
361:
357:
354:
351:
347:
344:
341:
338:
335:
331:
328:
324:
321:
318:
311:
308:
300:
299:
298:
296:
292:
288:
284:
283:PROPOSITION 1
279:
277:
273:
262:
255:
251:
247:
243:
239:
235:
230:
228:
224:
220:
216:
212:
208:
204:
200:
196:
192:
188:
184:
180:
176:
172:
168:
163:
161:
157:
153:
149:
145:
144:Venn diagrams
141:
137:
133:
129:
124:
122:
118:
114:
110:
106:
102:
98:
94:
91:) is assumed
90:
86:
82:
78:
75:and denoted
74:
70:
67:, called the
66:
62:
58:
54:
50:
49:universal set
45:
43:
39:
35:
31:
27:
23:
19:
1122:
1115:
1083:
1079:
1075:
1071:
1067:
1061:
1057:
1053:
1049:
1040:
1033:
1029:
1025:
1019:
1015:
1011:
1007:
1003:
997:
993:
989:
983:
979:
970:
968:
960:
956:
952:
948:
944:
940:
936:
929:
925:
921:
917:
913:
909:
905:
893:
885:
881:
877:
873:
869:
865:
861:
854:
850:
846:
842:
838:
834:
830:
826:
822:
815:
811:
807:
803:
799:
795:
791:
785:
781:
777:
773:
769:
765:
761:
755:
751:
747:
743:
739:
735:
731:
725:
721:
717:
713:
709:
705:
698:
694:
690:
686:
682:
678:
671:
660:
653:
645:
638:
631:
627:
623:
616:
612:
606:
602:
598:
594:
590:
583:
579:
575:
571:
565:
561:
557:
553:
547:
543:
539:
535:
529:
525:
521:
517:
508:
504:
500:
496:
494:
488:
484:
480:
474:
470:
466:
460:
450:
444:
440:
431:
427:
423:
417:
413:
407:
400:
394:
390:
383:
377:
373:
369:
363:
359:
355:
349:
345:
339:
333:
326:
322:
313:
306:
294:
290:
286:
282:
280:
275:
268:
257:
253:
249:
245:
241:
237:
231:
227:Zorn's lemma
164:
151:
147:
136:Huntington's
125:
108:
88:
84:
80:
76:
72:
64:
52:
46:
44:') of sets.
34:intersection
17:
15:
1118:5: 288-309.
1064: = {};
619: = {}
447: = {};
420: = {};
336: = {};
244:," denoted
165:Elementary
1106:References
955:) ∩ (
947:) = (
924:) ∪ (
916:) = (
880:) \ (
780:) ∩ (
772:) = (
750:) ∪ (
742:) = (
332:{} \
285:. For any
179:set theory
171:set theory
121:complement
55:, and the
51:, denoted
939: ∪ (
908: ∩ (
872: = (
868:) ∪
849: ∩ (
841:) \
833: = (
829:) ∩
810:) ∪(
794: \ (
764: \ (
734: \ (
720: ∪ (
712:) ∪
693: ∩ (
685:) ∩
69:power set
57:empty set
1133:Category
1093:See also
1078: ⊆
1074: ⊆
1070: ∩
1060: \
1052: ⊆
1039: ⊆
1028: ⊆
1018: =
1014: ∪
1006: ⊆
996: =
992: ∩
982: ⊆
959: ∪
951: ∪
943: ∩
928: ∩
920: ∩
912: ∪
845: =
716: =
689: =
630: =
626: \
615: ∩
546: ∪
542: =
538: ∪
528: ∩
524: =
520: ∩
487: =
483: ∪
473: =
469: ∩
459: =
443: \
430: =
426: \
416: \
376: =
372: ∪
362: =
358: ∩
162:axioms.
1080:A
410:= {}; *
274: ∩
267:and as
256: ∪
248: \
160:lattice
154:) is a
117:product
61:subsets
42:postfix
22:algebra
935:
904:
507:, and
225:, and
185:, and
132:axioms
93:closed
32:: ∪),
802:) = (
550:; * L
532:; * L
319:= {};
128:proof
101:model
26:union
963:). *
932:); *
894:The
728:); L
701:); L
597:) \
582:) =
564:) =
63:of
16:The
652:∪ (
586:; L
574:∩ (
568:; L
556:∪ (
397:; *
366:; *
352:; *
293:of
240:in
175:set
113:sum
103:of
99:or
83:).
71:of
24:of
1135::
1082:∪
898::
888:).
857:);
818:);
788:);
758:);
670:=
659:∪
644:∪
605:\
601:=
593:∪
578:∪
560:∩
511::
503:,
406:∩
393:=
389:∪
314:U'
305:=
301:{}
297::
278:.
229:.
221:,
217:,
213:,
209:,
205:,
201:,
197:,
193:,
181:,
177:,
119:,
115:,
1086:.
1084:B
1076:A
1072:B
1068:A
1062:B
1058:A
1054:B
1050:A
1046:;
1043:′
1041:A
1036:′
1034:B
1030:B
1026:A
1022:;
1020:B
1016:B
1012:A
1008:B
1004:A
1000:;
998:A
994:B
990:A
984:B
980:A
961:C
957:A
953:B
949:A
945:C
941:B
937:A
930:C
926:A
922:B
918:A
914:C
910:B
906:A
886:C
882:A
878:C
874:B
870:C
866:A
862:B
860:(
855:A
851:C
847:B
843:A
839:C
835:B
831:C
827:A
823:B
821:(
816:A
812:C
808:B
804:C
800:A
796:B
792:C
786:B
782:C
778:A
774:C
770:B
766:A
762:C
756:B
752:C
748:A
744:C
740:B
736:A
732:C
726:C
722:B
718:A
714:C
710:B
706:A
704:(
699:C
695:B
691:A
687:C
683:B
679:A
677:(
674:;
672:A
668:′
666:)
663:′
661:B
656:′
654:A
650:′
648:)
646:B
641:′
639:A
637:(
634:;
632:B
628:A
624:B
617:B
613:A
609:;
607:A
603:B
599:A
595:B
591:A
589:(
584:A
580:B
576:A
572:A
566:A
562:B
558:A
554:A
548:A
544:B
540:B
536:A
530:A
526:B
522:B
518:A
509:C
505:B
501:A
491:.
489:A
485:A
481:A
477:;
475:A
471:A
467:A
463:;
461:A
457:′
453:′
451:A
445:U
441:A
437:;
434:′
432:A
428:A
424:U
418:A
414:A
408:A
403:′
401:A
395:U
391:A
386:′
384:A
380:;
378:U
374:U
370:A
364:A
360:U
356:A
350:A
346:A
340:A
334:A
329:;
327:A
323:A
316:′
312:'
309:;
307:U
303:′
295:U
291:A
287:U
276:B
271:′
269:A
265:′
263:)
260:′
258:B
254:A
250:A
246:B
242:B
238:A
152:U
150:(
148:P
109:U
89:U
87:(
85:P
81:U
79:(
77:P
73:U
65:U
53:U
40:(
28:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.