1042:
is the identity function). However, families differ from sets in that the same object can appear multiple times with different indices in a family, whereas a set is a collection of distinct objects. A family contains any element exactly once
377:
2473:
so it is linearly dependent. The statement is therefore correct if it refers to the family of rows, but wrong if it refers to the set of rows. (The statement is also correct when "the rows" is interpreted as referring to a
2222:
is a collection of unordered distinct elements) and is linearly independent, but the family contains the same element twice (since indexed differently) and is linearly dependent (same vectors are linearly dependent).
1700:
Hence, by using a set instead of the family, some information might be lost. For example, an ordering on the index set of a family induces an ordering on the family, but no ordering on the corresponding image set.
1630:
2363:
2057:
1494:
928:
and conversely. Being an element of a family is equivalent to being in the range of the corresponding function. In practice, however, a family is viewed as a collection, rather than a function.
300:
1158:
2829:
1546:
1810:
1000:
794:
3025:
2929:
1755:
1100:
1376:
3079:
2629:
831:
if the index set is assumed to be known. Sometimes angle brackets or braces are used instead of parentheses, although the use of braces risks confusing indexed families with sets.
2975:
1983:
829:
2550:
2212:
1675:
926:
295:
2704:
2654:
2506:
1326:
1892:
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572:
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2104:
626:
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2130:
is defined as a property of a collection; it therefore is important if those vectors are linearly independent as a set or as a family. For example, if we consider
542:
446:
2154:
1406:
2764:
1698:
1433:
1201:
743:
2741:
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2570:
2301:
2277:
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2124:
2077:
1852:
1832:
1221:
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1040:
1020:
949:
878:
855:
718:
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646:
486:
466:
417:
397:
286:
266:
246:
214:
180:
156:
136:
112:
89:
138:
are referred to as making up the family. In this view, indexed families are interpreted as collections of indexed elements instead of functions. The set
2306:
3244:
1551:
61:, is a collection of real numbers, where a given function selects one real number for each integer (possibly the same) as indexing.
1996:
1438:
114:(that is, indexed families and mathematical functions are technically identical, just points of view are different). Often the
3249:
3038:
3224:
2934:
220:. For example, one could consider an uncountable family of subsets of the natural numbers indexed by the real numbers.
1105:
2804:
1499:
2401:
as a set is made of unique elements so it is linearly independent, but the matrix is not invertible as the matrix
1764:
954:
748:
3166:
3111:
3100:
3084:
2982:
2886:
1712:
1057:
1331:
2478:, in which the elements are also kept distinct but which lacks some of the structure of an indexed family.)
2591:
834:
289:
65:
1942:
799:
115:
2511:
2159:
1635:
2795:
1986:
883:
858:
69:
2684:
2634:
2127:
92:
3214: – Data structure used to hold a value that could take on several different, but fixed, types
2489:
1288:
3231:, 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM (volume).
3032:
1857:
1224:
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491:
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518:
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3154: – Data type that represents an ordered collection of elements (values or variables)
2133:
1381:
2746:
1680:
1415:
1183:
725:
372:{\displaystyle {\begin{aligned}f~:~&I\to X\\&i\mapsto x_{i}=f(i),\end{aligned}}}
3184:
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3172:
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141:
121:
97:
74:
28:
3238:
2303:
are linearly independent as a family, not as a set. For example, consider the matrix
217:
3211:
2871:
2581:
2106:
only makes sense with respect to this family, as sets are unordered so there is no
49:, is informally a collection of objects, each associated with an index from some
2883:
Index sets are often used in sums and other similar operations. For example, if
2402:
1409:
54:
38:
3160:
2931:
is an indexed family of numbers, the sum of all those numbers is denoted by
1048:
50:
17:
2409:
of the rows contains two elements indexed differently such as the 1st row
34:
Collection of objects, each associated with an element from some index set
3205:
3199: – family of objects whose definitions depend on a set of parameters
3139:
2709:
2475:
189:
3116:
2573:
58:
2631:
each element of the ordered pair is indexed by an element of the set
1625:{\displaystyle \left\{(-1)^{i}:i\in \mathbb {N} \right\}=\{-1,1\}.}
2720:
2659:
2585:
3169: – Indexed collection of objects and morphisms in a category
2358:{\displaystyle A={\begin{bmatrix}1&1\\1&1\end{bmatrix}}.}
837:
and indexed families are formally equivalent, since any function
2863:
indexing the matrix element at the 2nd row and the 5th column.
2283:
As in the previous example, it is important that the rows of
3187: – Manner of referring to elements of arrays or tensors
1342:
1111:
2052:{\displaystyle \left(v_{i}\right)_{i\in \{1,\ldots ,n\}}}
1489:{\displaystyle \left((-1)^{i}\right)_{i\in \mathbb {N} }}
3156:
Pages displaying short descriptions of redirect targets
2321:
3041:
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202:
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77:
3201:
Pages displaying wikidata descriptions as a fallback
3208: – Finite or infinite ordered list of elements
3181: – Any collection of sets, or subsets of a set
1677:does not carry information about any structures on
3073:
3019:
2969:
2923:
2855:
2823:
2787:
2758:
2735:
2698:
2671:
2648:
2623:
2588:) is a family indexed by the set of two elements,
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208:
174:
150:
130:
106:
83:
3119:giving rise to an indexed family of objects in a
3193: – A generalization of a sequence of points
1153:{\displaystyle {\mathcal {X}}=\{x_{i}:i\in I\},}
2831:which elements are ordered pairs; for example,
2824:{\displaystyle \mathbf {n} \times \mathbf {m} }
2233:
1937:
1935:For example, consider the following sentence:
1541:{\displaystyle \mathbb {N} =\{1,2,3,\ldots \}}
8:
2615:
2603:
2536:
2515:
2044:
2026:
1805:{\displaystyle \left(A_{i}\right)_{i\in I},}
1664:
1639:
1616:
1601:
1535:
1511:
1144:
1119:
995:{\displaystyle \left(x_{x}\right)_{x\in X},}
789:{\displaystyle \left(x_{i}\right)_{i\in I},}
3020:{\displaystyle \left(A_{i}\right)_{i\in I}}
2924:{\displaystyle \left(a_{i}\right)_{i\in I}}
2218:of them consists of only one element (as a
1750:{\displaystyle \left(B_{i}\right)_{i\in J}}
1095:{\displaystyle \left(x_{i}\right)_{i\in I}}
3175: – In mathematics, operation on sets
3062:
3046:
3040:
3005:
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2984:
2958:
2942:
2936:
2909:
2899:
2888:
2836:
2816:
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2728:
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2638:
2636:
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2513:
2493:
2491:
2446:
2414:
2374:
2369:of the rows consists of a single element
2316:
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2240:
2180:
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2111:
2090:
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2019:
2009:
1998:
1969:
1950:
1944:
1899:
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1839:
1819:
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1735:
1725:
1714:
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1637:
1589:
1588:
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1553:
1504:
1503:
1501:
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1440:
1417:
1393:
1385:
1383:
1363:
1355:
1347:
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1340:
1335:
1333:
1309:
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682:
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341:
299:
297:
273:
253:
233:
201:
167:
143:
123:
99:
76:
1371:{\displaystyle |{\mathcal {X}}|\leq |I|}
3163: – Category-theoretic construction
3074:{\displaystyle \bigcup _{i\in I}A_{i}.}
3229:Encyclopedic Dictionary of Mathematics
64:More formally, an indexed family is a
2624:{\displaystyle \mathbf {2} =\{1,2\};}
2231:Suppose a text states the following:
7:
2970:{\displaystyle \sum _{i\in I}a_{i}.}
192:are one type of families indexed by
1978:{\displaystyle v_{1},\ldots ,v_{n}}
1022:is indexed by itself (meaning that
2126:-th vector of a set. Furthermore,
824:{\displaystyle \left(x_{i}\right)}
25:
2545:{\displaystyle \{1,2,\ldots n\},}
2207:{\displaystyle v_{1}=v_{2}=(1,0)}
2059:denotes a family of vectors. The
3035:of all those sets is denoted by
2817:
2809:
2689:
2639:
2596:
2494:
1670:{\displaystyle \{x_{i}:i\in I\}}
2681:is a family indexed by the set
1496:indexed by the natural numbers
921:{\displaystyle (f(i))_{i\in I}}
3130:, indexed by another category
2879:Operations on indexed families
2850:
2838:
2460:
2448:
2428:
2416:
2405:is 0. On the other hands, the
2388:
2376:
2201:
2189:
1570:
1560:
1458:
1448:
1394:
1386:
1364:
1356:
1348:
1336:
1047:the corresponding function is
903:
899:
893:
887:
531:
525:
435:
429:
359:
353:
334:
320:
1:
3225:Mathematical Society of Japan
2699:{\displaystyle \mathbf {n} .}
2649:{\displaystyle \mathbf {2} .}
2214:as the same vector, then the
3245:Basic concepts in set theory
2501:{\displaystyle \mathbf {n} }
1321:{\displaystyle x_{i}=x_{j}.}
196:. In general, the index set
2798:is a family indexed by the
2712:is a family indexed by the
1887:{\displaystyle B_{i}=A_{i}}
53:. For example, a family of
3266:
3142:depending on two indices.
3098:
2743:-tuple for an unspecified
1435:For example, the sequence
26:
3167:Diagram (category theory)
3105:The analogous concept in
3101:Diagram (category theory)
2870:is a family indexed by a
2788:{\displaystyle n\times m}
2279:are linearly independent.
601:is used to indicate that
3095:Usage in category theory
2766:or an infinite sequence.
1252:{\displaystyle i,j\in I}
216:is not restricted to be
57:, indexed by the set of
27:Not to be confused with
1916:{\displaystyle i\in J.}
1278:{\displaystyle i\neq j}
951:gives rise to a family
670:{\displaystyle i\in I.}
3075:
3021:
2971:
2925:
2857:
2825:
2789:
2760:
2737:
2700:
2673:
2650:
2625:
2566:
2546:
2502:
2467:
2435:
2395:
2359:
2297:
2281:
2273:
2249:
2208:
2150:
2120:
2100:
2073:
2053:
1991:
1979:
1917:
1888:
1848:
1828:
1806:
1751:
1694:
1671:
1626:
1542:
1490:
1429:
1402:
1372:
1322:
1279:
1253:
1223:is not required to be
1217:
1197:
1174:
1160:that is, the image of
1154:
1096:
1036:
1016:
996:
945:
922:
874:
851:
825:
790:
739:
714:
691:
671:
642:
622:
595:
568:
567:{\displaystyle x_{3}.}
538:
509:
482:
462:
442:
413:
393:
373:
282:
262:
242:
210:
176:
152:
132:
108:
85:
3250:Mathematical notation
3076:
3022:
2972:
2926:
2858:
2856:{\displaystyle (2,5)}
2826:
2790:
2761:
2738:
2701:
2674:
2651:
2626:
2567:
2547:
2503:
2468:
2466:{\displaystyle (1,1)}
2436:
2434:{\displaystyle (1,1)}
2396:
2394:{\displaystyle (1,1)}
2360:
2298:
2274:
2250:
2209:
2151:
2121:
2101:
2099:{\displaystyle v_{i}}
2074:
2054:
1980:
1918:
1889:
1849:
1829:
1807:
1761:of an indexed family
1752:
1695:
1672:
1632:In addition, the set
1627:
1543:
1491:
1430:
1403:
1373:
1323:
1280:
1254:
1218:
1198:
1175:
1155:
1097:
1037:
1017:
997:
946:
923:
875:
852:
826:
791:
740:
715:
699:family of elements in
692:
672:
643:
623:
621:{\displaystyle x_{i}}
596:
594:{\displaystyle x_{i}}
569:
539:
510:
508:{\displaystyle x_{i}}
483:
463:
443:
414:
394:
374:
283:
263:
243:
211:
177:
153:
133:
109:
86:
66:mathematical function
3039:
2983:
2935:
2887:
2835:
2805:
2773:
2747:
2727:
2685:
2663:
2635:
2592:
2556:
2512:
2490:
2445:
2413:
2373:
2307:
2287:
2263:
2239:
2160:
2134:
2110:
2083:
2063:
1997:
1987:linearly independent
1943:
1898:
1858:
1838:
1818:
1765:
1713:
1681:
1636:
1552:
1500:
1439:
1416:
1382:
1332:
1289:
1263:
1231:
1207:
1184:
1164:
1106:
1058:
1026:
1006:
955:
935:
884:
864:
841:
800:
749:
745:which is denoted by
726:
704:
681:
652:
632:
605:
578:
548:
537:{\displaystyle f(3)}
519:
492:
472:
452:
441:{\displaystyle f(i)}
423:
403:
383:
296:
272:
252:
232:
200:
166:
142:
122:
98:
75:
2149:{\displaystyle n=2}
2128:linear independence
1401:{\displaystyle |A|}
697:thus establishes a
468:under the function
162:of the family, and
3089:Cartesian products
3071:
3057:
3017:
2967:
2953:
2921:
2853:
2821:
2785:
2759:{\displaystyle n,}
2756:
2733:
2696:
2669:
2646:
2621:
2562:
2542:
2508:be the finite set
2498:
2463:
2431:
2391:
2355:
2346:
2293:
2269:
2245:
2204:
2146:
2116:
2096:
2069:
2049:
1975:
1913:
1884:
1844:
1824:
1802:
1747:
1709:An indexed family
1693:{\displaystyle I.}
1690:
1667:
1622:
1538:
1486:
1428:{\displaystyle A.}
1425:
1398:
1368:
1318:
1275:
1249:
1227:, there may exist
1213:
1203:Since the mapping
1196:{\displaystyle f.}
1193:
1170:
1150:
1092:
1054:An indexed family
1032:
1012:
992:
941:
918:
870:
847:
821:
786:
738:{\displaystyle I,}
735:
710:
687:
667:
638:
628:is the element of
618:
591:
564:
534:
505:
478:
458:
438:
409:
389:
369:
367:
278:
258:
238:
206:
172:
148:
128:
104:
81:
68:together with its
3197:Parametric family
3191:Net (mathematics)
3138:, and related by
3115:. A diagram is a
3042:
2938:
2800:Cartesian product
2736:{\displaystyle n}
2672:{\displaystyle n}
2565:{\displaystyle n}
2296:{\displaystyle A}
2272:{\displaystyle A}
2248:{\displaystyle A}
2119:{\displaystyle i}
2072:{\displaystyle i}
1847:{\displaystyle I}
1827:{\displaystyle J}
1705:Indexed subfamily
1216:{\displaystyle f}
1173:{\displaystyle I}
1035:{\displaystyle f}
1015:{\displaystyle X}
944:{\displaystyle X}
880:induces a family
873:{\displaystyle I}
850:{\displaystyle f}
713:{\displaystyle X}
690:{\displaystyle f}
641:{\displaystyle X}
481:{\displaystyle f}
461:{\displaystyle i}
412:{\displaystyle I}
399:is an element of
392:{\displaystyle i}
314:
308:
281:{\displaystyle f}
261:{\displaystyle X}
241:{\displaystyle I}
224:Formal definition
209:{\displaystyle I}
175:{\displaystyle X}
151:{\displaystyle I}
131:{\displaystyle X}
107:{\displaystyle X}
84:{\displaystyle I}
16:(Redirected from
3257:
3202:
3157:
3137:
3129:
3080:
3078:
3077:
3072:
3067:
3066:
3056:
3026:
3024:
3023:
3018:
3016:
3015:
3004:
3000:
2999:
2976:
2974:
2973:
2968:
2963:
2962:
2952:
2930:
2928:
2927:
2922:
2920:
2919:
2908:
2904:
2903:
2862:
2860:
2859:
2854:
2830:
2828:
2827:
2822:
2820:
2812:
2794:
2792:
2791:
2786:
2765:
2763:
2762:
2757:
2742:
2740:
2739:
2734:
2705:
2703:
2702:
2697:
2692:
2678:
2676:
2675:
2670:
2655:
2653:
2652:
2647:
2642:
2630:
2628:
2627:
2622:
2599:
2571:
2569:
2568:
2563:
2551:
2549:
2548:
2543:
2507:
2505:
2504:
2499:
2497:
2472:
2470:
2469:
2464:
2441:and the 2nd row
2440:
2438:
2437:
2432:
2400:
2398:
2397:
2392:
2364:
2362:
2361:
2356:
2351:
2350:
2302:
2300:
2299:
2294:
2278:
2276:
2275:
2270:
2254:
2252:
2251:
2246:
2235:A square matrix
2213:
2211:
2210:
2205:
2185:
2184:
2172:
2171:
2155:
2153:
2152:
2147:
2125:
2123:
2122:
2117:
2105:
2103:
2102:
2097:
2095:
2094:
2078:
2076:
2075:
2070:
2058:
2056:
2055:
2050:
2048:
2047:
2018:
2014:
2013:
1984:
1982:
1981:
1976:
1974:
1973:
1955:
1954:
1922:
1920:
1919:
1914:
1893:
1891:
1890:
1885:
1883:
1882:
1870:
1869:
1853:
1851:
1850:
1845:
1833:
1831:
1830:
1825:
1811:
1809:
1808:
1803:
1798:
1797:
1786:
1782:
1781:
1756:
1754:
1753:
1748:
1746:
1745:
1734:
1730:
1729:
1699:
1697:
1696:
1691:
1676:
1674:
1673:
1668:
1651:
1650:
1631:
1629:
1628:
1623:
1597:
1593:
1592:
1578:
1577:
1547:
1545:
1544:
1539:
1507:
1495:
1493:
1492:
1487:
1485:
1484:
1483:
1471:
1467:
1466:
1465:
1434:
1432:
1431:
1426:
1407:
1405:
1404:
1399:
1397:
1389:
1377:
1375:
1374:
1369:
1367:
1359:
1351:
1346:
1345:
1339:
1327:
1325:
1324:
1319:
1314:
1313:
1301:
1300:
1284:
1282:
1281:
1276:
1258:
1256:
1255:
1250:
1222:
1220:
1219:
1214:
1202:
1200:
1199:
1194:
1179:
1177:
1176:
1171:
1159:
1157:
1156:
1151:
1131:
1130:
1115:
1114:
1101:
1099:
1098:
1093:
1091:
1090:
1079:
1075:
1074:
1041:
1039:
1038:
1033:
1021:
1019:
1018:
1013:
1001:
999:
998:
993:
988:
987:
976:
972:
971:
950:
948:
947:
942:
927:
925:
924:
919:
917:
916:
879:
877:
876:
871:
856:
854:
853:
848:
830:
828:
827:
822:
820:
816:
815:
795:
793:
792:
787:
782:
781:
770:
766:
765:
744:
742:
741:
736:
719:
717:
716:
711:
696:
694:
693:
688:
676:
674:
673:
668:
647:
645:
644:
639:
627:
625:
624:
619:
617:
616:
600:
598:
597:
592:
590:
589:
573:
571:
570:
565:
560:
559:
543:
541:
540:
535:
514:
512:
511:
506:
504:
503:
487:
485:
484:
479:
467:
465:
464:
459:
447:
445:
444:
439:
418:
416:
415:
410:
398:
396:
395:
390:
378:
376:
375:
370:
368:
346:
345:
329:
312:
306:
287:
285:
284:
279:
267:
265:
264:
259:
247:
245:
244:
239:
215:
213:
212:
207:
181:
179:
178:
173:
157:
155:
154:
149:
137:
135:
134:
129:
113:
111:
110:
105:
90:
88:
87:
82:
21:
3265:
3264:
3260:
3259:
3258:
3256:
3255:
3254:
3235:
3234:
3221:
3200:
3155:
3152:Array data type
3148:
3131:
3123:
3107:category theory
3103:
3097:
3058:
3037:
3036:
2991:
2987:
2986:
2981:
2980:
2954:
2933:
2932:
2895:
2891:
2890:
2885:
2884:
2881:
2833:
2832:
2803:
2802:
2771:
2770:
2745:
2744:
2725:
2724:
2714:natural numbers
2683:
2682:
2661:
2660:
2633:
2632:
2590:
2589:
2554:
2553:
2510:
2509:
2488:
2487:
2484:
2443:
2442:
2411:
2410:
2371:
2370:
2345:
2344:
2339:
2333:
2332:
2327:
2317:
2305:
2304:
2285:
2284:
2261:
2260:
2255:is invertible,
2237:
2236:
2229:
2176:
2163:
2158:
2157:
2132:
2131:
2108:
2107:
2086:
2081:
2080:
2061:
2060:
2005:
2001:
2000:
1995:
1994:
1965:
1946:
1941:
1940:
1933:
1931:Indexed vectors
1928:
1896:
1895:
1874:
1861:
1856:
1855:
1836:
1835:
1834:is a subset of
1816:
1815:
1773:
1769:
1768:
1763:
1762:
1721:
1717:
1716:
1711:
1710:
1707:
1679:
1678:
1642:
1634:
1633:
1569:
1559:
1555:
1550:
1549:
1498:
1497:
1457:
1447:
1443:
1442:
1437:
1436:
1414:
1413:
1380:
1379:
1330:
1329:
1305:
1292:
1287:
1286:
1261:
1260:
1229:
1228:
1205:
1204:
1182:
1181:
1162:
1161:
1122:
1104:
1103:
1066:
1062:
1061:
1056:
1055:
1024:
1023:
1004:
1003:
963:
959:
958:
953:
952:
933:
932:
902:
882:
881:
862:
861:
839:
838:
807:
803:
798:
797:
757:
753:
752:
747:
746:
724:
723:
702:
701:
679:
678:
650:
649:
630:
629:
608:
603:
602:
581:
576:
575:
551:
546:
545:
517:
516:
515:. For example,
495:
490:
489:
470:
469:
450:
449:
421:
420:
401:
400:
381:
380:
366:
365:
337:
327:
326:
315:
294:
293:
270:
269:
250:
249:
230:
229:
226:
198:
197:
194:natural numbers
164:
163:
140:
139:
120:
119:
96:
95:
73:
72:
35:
32:
23:
22:
15:
12:
11:
5:
3263:
3261:
3253:
3252:
3247:
3237:
3236:
3233:
3232:
3220:
3217:
3216:
3215:
3209:
3203:
3194:
3188:
3185:Index notation
3182:
3179:Family of sets
3176:
3173:Disjoint union
3170:
3164:
3158:
3147:
3144:
3099:Main article:
3096:
3093:
3070:
3065:
3061:
3055:
3052:
3049:
3045:
3029:family of sets
3014:
3011:
3008:
3003:
2998:
2994:
2990:
2966:
2961:
2957:
2951:
2948:
2945:
2941:
2918:
2915:
2912:
2907:
2902:
2898:
2894:
2880:
2877:
2876:
2875:
2864:
2852:
2849:
2846:
2843:
2840:
2819:
2815:
2811:
2784:
2781:
2778:
2767:
2755:
2752:
2732:
2717:
2706:
2695:
2691:
2668:
2656:
2645:
2641:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2598:
2572:is a positive
2561:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2496:
2483:
2482:Other examples
2480:
2462:
2459:
2456:
2453:
2450:
2430:
2427:
2424:
2421:
2418:
2390:
2387:
2384:
2381:
2378:
2354:
2349:
2343:
2340:
2338:
2335:
2334:
2331:
2328:
2326:
2323:
2322:
2320:
2315:
2312:
2292:
2268:
2257:if and only if
2244:
2228:
2225:
2203:
2200:
2197:
2194:
2191:
2188:
2183:
2179:
2175:
2170:
2166:
2145:
2142:
2139:
2115:
2093:
2089:
2068:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2017:
2012:
2008:
2004:
1972:
1968:
1964:
1961:
1958:
1953:
1949:
1932:
1929:
1927:
1924:
1912:
1909:
1906:
1903:
1894:holds for all
1881:
1877:
1873:
1868:
1864:
1843:
1823:
1813:if and only if
1801:
1796:
1793:
1790:
1785:
1780:
1776:
1772:
1744:
1741:
1738:
1733:
1728:
1724:
1720:
1706:
1703:
1689:
1686:
1666:
1663:
1660:
1657:
1654:
1649:
1645:
1641:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1596:
1591:
1587:
1584:
1581:
1576:
1572:
1568:
1565:
1562:
1558:
1548:has image set
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1506:
1482:
1478:
1475:
1470:
1464:
1460:
1456:
1453:
1450:
1446:
1424:
1421:
1396:
1392:
1388:
1366:
1362:
1358:
1354:
1350:
1344:
1338:
1317:
1312:
1308:
1304:
1299:
1295:
1274:
1271:
1268:
1248:
1245:
1242:
1239:
1236:
1212:
1192:
1189:
1169:
1149:
1146:
1143:
1140:
1137:
1134:
1129:
1125:
1121:
1118:
1113:
1102:defines a set
1089:
1086:
1083:
1078:
1073:
1069:
1065:
1045:if and only if
1031:
1011:
991:
986:
983:
980:
975:
970:
966:
962:
940:
915:
912:
909:
905:
901:
898:
895:
892:
889:
869:
846:
819:
814:
810:
806:
785:
780:
777:
774:
769:
764:
760:
756:
734:
731:
709:
686:
666:
663:
660:
657:
637:
615:
611:
588:
584:
563:
558:
554:
544:is denoted by
533:
530:
527:
524:
502:
498:
488:is denoted by
477:
457:
437:
434:
431:
428:
419:and the image
408:
388:
364:
361:
358:
355:
352:
349:
344:
340:
336:
333:
330:
328:
325:
322:
319:
316:
311:
305:
302:
301:
277:
257:
237:
225:
222:
205:
171:
158:is called the
147:
127:
103:
80:
47:indexed family
33:
29:Family of sets
24:
14:
13:
10:
9:
6:
4:
3:
2:
3262:
3251:
3248:
3246:
3243:
3242:
3240:
3230:
3226:
3223:
3222:
3218:
3213:
3210:
3207:
3204:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3153:
3150:
3149:
3145:
3143:
3141:
3136:
3135:
3128:
3127:
3122:
3118:
3114:
3113:
3108:
3102:
3094:
3092:
3090:
3086:
3085:intersections
3083:Likewise for
3081:
3068:
3063:
3059:
3053:
3050:
3047:
3043:
3034:
3030:
3012:
3009:
3006:
3001:
2996:
2992:
2988:
2977:
2964:
2959:
2955:
2949:
2946:
2943:
2939:
2916:
2913:
2910:
2905:
2900:
2896:
2892:
2878:
2873:
2869:
2865:
2847:
2844:
2841:
2813:
2801:
2797:
2782:
2779:
2776:
2768:
2753:
2750:
2730:
2722:
2718:
2715:
2711:
2707:
2693:
2680:
2666:
2657:
2643:
2618:
2612:
2609:
2606:
2600:
2587:
2583:
2579:
2578:
2577:
2575:
2559:
2539:
2533:
2530:
2527:
2524:
2521:
2518:
2481:
2479:
2477:
2457:
2454:
2451:
2425:
2422:
2419:
2408:
2404:
2385:
2382:
2379:
2368:
2352:
2347:
2341:
2336:
2329:
2324:
2318:
2313:
2310:
2290:
2280:
2266:
2258:
2242:
2232:
2226:
2224:
2221:
2217:
2198:
2195:
2192:
2186:
2181:
2177:
2173:
2168:
2164:
2143:
2140:
2137:
2129:
2113:
2091:
2087:
2066:
2041:
2038:
2035:
2032:
2029:
2023:
2020:
2015:
2010:
2006:
2002:
1990:
1988:
1970:
1966:
1962:
1959:
1956:
1951:
1947:
1936:
1930:
1925:
1923:
1910:
1907:
1904:
1901:
1879:
1875:
1871:
1866:
1862:
1841:
1821:
1814:
1799:
1794:
1791:
1788:
1783:
1778:
1774:
1770:
1760:
1742:
1739:
1736:
1731:
1726:
1722:
1718:
1704:
1702:
1687:
1684:
1661:
1658:
1655:
1652:
1647:
1643:
1619:
1613:
1610:
1607:
1604:
1598:
1594:
1585:
1582:
1579:
1574:
1566:
1563:
1556:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1508:
1476:
1473:
1468:
1462:
1454:
1451:
1444:
1422:
1419:
1411:
1390:
1360:
1352:
1315:
1310:
1306:
1302:
1297:
1293:
1272:
1269:
1266:
1246:
1243:
1240:
1237:
1234:
1226:
1210:
1190:
1187:
1167:
1147:
1141:
1138:
1135:
1132:
1127:
1123:
1116:
1087:
1084:
1081:
1076:
1071:
1067:
1063:
1052:
1050:
1046:
1029:
1009:
989:
984:
981:
978:
973:
968:
964:
960:
938:
929:
913:
910:
907:
896:
890:
867:
860:
844:
836:
832:
817:
812:
808:
804:
783:
778:
775:
772:
767:
762:
758:
754:
732:
729:
722:
707:
700:
684:
677:The function
664:
661:
658:
655:
635:
613:
609:
586:
582:
561:
556:
552:
528:
522:
500:
496:
475:
455:
432:
426:
406:
386:
362:
356:
350:
347:
342:
338:
331:
323:
317:
309:
303:
291:
275:
255:
235:
223:
221:
219:
203:
195:
191:
187:
185:
169:
161:
145:
125:
117:
101:
94:
78:
71:
67:
62:
60:
56:
52:
48:
44:
40:
30:
19:
3228:
3212:Tagged union
3133:
3132:
3125:
3124:
3110:
3109:is called a
3104:
3082:
2978:
2882:
2872:directed set
2708:An infinite
2582:ordered pair
2485:
2406:
2366:
2282:
2259:the rows of
2234:
2230:
2215:
1992:
1939:The vectors
1938:
1934:
1758:
1708:
1408:denotes the
1053:
930:
833:
720:
698:
268:be sets and
227:
188:
183:
159:
63:
55:real numbers
46:
42:
36:
2403:determinant
2079:-th vector
1412:of the set
1410:cardinality
648:indexed by
574:The symbol
184:indexed set
118:of the set
39:mathematics
18:Indexed set
3239:Categories
3219:References
1285:such that
796:or simply
721:indexed by
292:such that
3161:Coproduct
3140:morphisms
3051:∈
3044:⋃
3010:∈
2947:∈
2940:∑
2914:∈
2814:×
2780:×
2531:…
2036:…
2024:∈
1960:…
1905:∈
1792:∈
1759:subfamily
1740:∈
1659:∈
1605:−
1586:∈
1564:−
1533:…
1477:∈
1452:−
1353:≤
1270:≠
1244:∈
1225:injective
1139:∈
1085:∈
1049:injective
982:∈
911:∈
835:Functions
776:∈
659:∈
335:↦
321:→
218:countable
190:Sequences
160:index set
51:index set
3206:Sequence
3146:See also
3121:category
2710:sequence
2476:multiset
2227:Matrices
1926:Examples
1378:, where
931:Any set
290:function
116:elements
59:integers
3117:functor
3112:diagram
2574:integer
857:with a
182:is the
3031:, the
2796:matrix
2723:is an
2679:-tuple
2552:where
2407:family
1328:Thus,
1180:under
1002:where
859:domain
379:where
313:
307:
70:domain
43:family
3033:union
3027:is a
2979:When
2586:tuple
1993:Here
1757:is a
1259:with
93:image
45:, or
3087:and
2721:list
2486:Let
2365:The
2156:and
1985:are
1854:and
248:and
228:Let
91:and
41:, a
2868:net
2769:An
2658:An
2584:(2-
2580:An
2367:set
2220:set
2216:set
448:of
37:In
3241::
3227:,
3091:.
2866:A
2719:A
2576:.
1051:.
288:a
186:.
3134:J
3126:C
3069:.
3064:i
3060:A
3054:I
3048:i
3013:I
3007:i
3002:)
2997:i
2993:A
2989:(
2965:.
2960:i
2956:a
2950:I
2944:i
2917:I
2911:i
2906:)
2901:i
2897:a
2893:(
2874:.
2851:)
2848:5
2845:,
2842:2
2839:(
2818:m
2810:n
2783:m
2777:n
2754:,
2751:n
2731:n
2716:.
2694:.
2690:n
2667:n
2644:.
2640:2
2619:;
2616:}
2613:2
2610:,
2607:1
2604:{
2601:=
2597:2
2560:n
2540:,
2537:}
2534:n
2528:,
2525:2
2522:,
2519:1
2516:{
2495:n
2461:)
2458:1
2455:,
2452:1
2449:(
2429:)
2426:1
2423:,
2420:1
2417:(
2389:)
2386:1
2383:,
2380:1
2377:(
2353:.
2348:]
2342:1
2337:1
2330:1
2325:1
2319:[
2314:=
2311:A
2291:A
2267:A
2243:A
2202:)
2199:0
2196:,
2193:1
2190:(
2187:=
2182:2
2178:v
2174:=
2169:1
2165:v
2144:2
2141:=
2138:n
2114:i
2092:i
2088:v
2067:i
2045:}
2042:n
2039:,
2033:,
2030:1
2027:{
2021:i
2016:)
2011:i
2007:v
2003:(
1989:.
1971:n
1967:v
1963:,
1957:,
1952:1
1948:v
1911:.
1908:J
1902:i
1880:i
1876:A
1872:=
1867:i
1863:B
1842:I
1822:J
1800:,
1795:I
1789:i
1784:)
1779:i
1775:A
1771:(
1743:J
1737:i
1732:)
1727:i
1723:B
1719:(
1688:.
1685:I
1665:}
1662:I
1656:i
1653::
1648:i
1644:x
1640:{
1620:.
1617:}
1614:1
1611:,
1608:1
1602:{
1599:=
1595:}
1590:N
1583:i
1580::
1575:i
1571:)
1567:1
1561:(
1557:{
1536:}
1530:,
1527:3
1524:,
1521:2
1518:,
1515:1
1512:{
1509:=
1505:N
1481:N
1474:i
1469:)
1463:i
1459:)
1455:1
1449:(
1445:(
1423:.
1420:A
1395:|
1391:A
1387:|
1365:|
1361:I
1357:|
1349:|
1343:X
1337:|
1316:.
1311:j
1307:x
1303:=
1298:i
1294:x
1273:j
1267:i
1247:I
1241:j
1238:,
1235:i
1211:f
1191:.
1188:f
1168:I
1148:,
1145:}
1142:I
1136:i
1133::
1128:i
1124:x
1120:{
1117:=
1112:X
1088:I
1082:i
1077:)
1072:i
1068:x
1064:(
1030:f
1010:X
990:,
985:X
979:x
974:)
969:x
965:x
961:(
939:X
914:I
908:i
904:)
900:)
897:i
894:(
891:f
888:(
868:I
845:f
818:)
813:i
809:x
805:(
784:,
779:I
773:i
768:)
763:i
759:x
755:(
733:,
730:I
708:X
685:f
665:.
662:I
656:i
636:X
614:i
610:x
587:i
583:x
562:.
557:3
553:x
532:)
529:3
526:(
523:f
501:i
497:x
476:f
456:i
436:)
433:i
430:(
427:f
407:I
387:i
363:,
360:)
357:i
354:(
351:f
348:=
343:i
339:x
332:i
324:X
318:I
310::
304:f
276:f
256:X
236:I
204:I
170:X
146:I
126:X
102:X
79:I
31:.
20:)
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