137:(while possibly minimizing errors and confusion at the same time). However, since the concept of formal/syntactical correctness depends on both time and context, certain notations in mathematics that are flagged as abuse in one context could be formally correct in one or more other contexts. Time-dependent abuses of notation may occur when novel notations are introduced to a theory some time before the theory is first formalized; these may be formally corrected by solidifying and/or otherwise improving the theory.
25:
329:. In general, there is no problem with this if the object under reference is well understood, and avoiding such an abuse of notation might even make mathematical texts more pedantic and more difficult to read. When this abuse of notation may be confusing, one may distinguish between these structures by denoting
499:— even though they are technically distinct mathematical objects. Nevertheless, it could occur on some occasions that two different topologies are considered simultaneously on the same set. In which case, one must exercise care and use notation such as
1443:} is denoted . But in practice, if the remainder of the discussion is focused on the equivalence classes rather than the individual elements of the underlying set, then it is common to drop the square brackets in the discussion.
1077:
1390:
Another example of similar abuses occurs in statements such as "there are two non-Abelian groups of order 8", which more strictly stated means "there are two isomorphism classes of non-Abelian groups of order 8".
985:). Once this desired property is defined, there may be various ways to construct the structure, and the corresponding results are formally different objects, but which have exactly the same properties (i.e.,
574:
401:
1233:
971:
533:
1377:
361:
244:
441:
497:
782:
307:
906:
863:
824:
1309:
1271:
164:— rather than a notation — is misused. Abuse of language is an almost synonymous expression for abuses that are non-notational by nature. For example, while the word
1159:
1133:
1107:
784:
be a function ..." This abuse of notation is widely used, as it simplifies the formulation, and the systematic use of a correct notation quickly becomes pedantic.
667:
611:
989:). As there is no way to distinguish these isomorphic objects through their properties, it is standard to consider them as equal, even if this is formally wrong.
638:
926:
747:
727:
707:
687:
1002:
283:. It is a common abuse of notation to use the same notation for the underlying set and the structured object (a phenomenon known as
89:
61:
68:
145:
of notation, which does not have the presentational benefits of the former and should be avoided (such as the misuse of
42:
1481:)". The elements of that group would then be , , ..., , but in practice they are usually denoted simply as 0, 1, ...,
1669:
108:
133:
in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct
75:
1664:
46:
57:
1634:
366:
205:". Another common abuse of language consists in identifying two mathematical objects that are different, but
973:
Nevertheless, this abuse of notation is widely used, since it is more concise but generally not confusing.
1164:
931:
538:
251:
502:
1314:
332:
220:
419:
478:
752:
146:
290:
35:
1535:
872:
829:
790:
614:
276:
134:
82:
1555:
1276:
1238:
206:
130:
1138:
1112:
1086:
1580:
1455:
1404:
182:
165:
643:
587:
8:
1384:
314:
268:
620:
1447:
1416:
982:
911:
732:
712:
692:
672:
322:
171:
1379:
would mean nothing. However, these equalities can be legitimized and made rigorous in
787:
A similar abuse of notation occurs in sentences such as "Let us consider the function
275:, often called the underlying set, equipped with some additional structure, such as a
1497:
1493:
1400:
993:
407:
280:
272:
210:
981:
Many mathematical structures are defined through a characterizing property (often a
1607:
1523:
214:
1508:
The terms "abuse of language" and "abuse of notation" depend on context. Writing "
1380:
247:
326:
1658:
1489:
1451:
194:
175:
1488:
Another example is the space of (classes of) measurable functions over a
1461:
can be formed by partitioning the integers via the equivalence relation "
1072:{\displaystyle (E\times F)\times G=E\times (F\times G)=E\times F\times G}
122:
986:
866:
613:
be a function ...". This is an abuse of notation, as the name of the
24:
1560:
1543:
454:
318:
310:
869:
expression, not a function per se. The function that associates
444:
1635:"More about the languages of math — Suppression of parameters"
709:
of its domain. More precisely correct phrasings include "Let
584:
One may encounter, in many textbooks, sentences such as "Let
576:
to distinguish between the different topological spaces.
1411:
instead of is an abuse of notation. Formally, if a set
1496:
functions, where the equivalence relation is equality "
469:
as both the underlying set, and the pair consisting of
457:). Most frequently, one considers only one topology on
217:
with the name of its underlying set, or identifying to
1317:
1279:
1241:
1167:
1141:
1115:
1089:
1005:
934:
914:
875:
832:
793:
755:
735:
715:
695:
675:
646:
623:
590:
541:
505:
481:
422:
369:
335:
293:
223:
1534:
is almost always an abuse of notation, but not in a
49:. Unsourced material may be challenged and removed.
1371:
1303:
1265:
1227:
1153:
1127:
1101:
1071:
965:
920:
900:
857:
818:
776:
741:
721:
701:
681:
661:
632:
605:
568:
527:
491:
435:
395:
355:
301:
262:
238:
1656:
1546:in the category of sets and partial functions.
463:, so there is usually no problem in referring
1419:by an equivalence relation ~, then for each
1083:But this is strictly speaking not true: if
976:
213:with its value, identifying a group with a
374:
363:the group of integers with addition, and
340:
295:
226:
109:Learn how and when to remove this message
396:{\displaystyle (\mathbb {Z} ,+,\cdot )}
209:. Other examples include identifying a
1657:
1394:
996:, which is often seen as associative:
16:Informal use of mathematical notation
1629:
1627:
1602:
1600:
579:
416:(the underlying set) and a topology
47:adding citations to reliable sources
18:
1228:{\displaystyle ((x,y),z)=(x,(y,z))}
966:{\displaystyle x\mapsto x^{2}+x+1.}
569:{\displaystyle (X,{\mathcal {T}}')}
443:which is characterized by a set of
250:of dimension three equipped with a
13:
554:
528:{\displaystyle (X,{\mathcal {T}})}
517:
484:
425:
14:
1681:
1624:
1597:
1372:{\displaystyle ((x,y),z)=(x,y,z)}
356:{\displaystyle (\mathbb {Z} ,+)}
239:{\displaystyle \mathbb {R} ^{3}}
23:
1581:"Common Errors in College Math"
1503:
436:{\displaystyle {\mathcal {T}},}
263:Structured mathematical objects
34:needs additional citations for
1608:"Glossary — Abuse of notation"
1573:
1366:
1348:
1342:
1333:
1321:
1318:
1298:
1286:
1254:
1242:
1222:
1219:
1207:
1198:
1192:
1183:
1171:
1168:
1048:
1036:
1018:
1006:
938:
771:
765:
759:
729:be a function of the variable
656:
650:
600:
594:
563:
542:
522:
506:
492:{\displaystyle {\mathcal {T}}}
390:
370:
350:
336:
325:of integers with addition and
1:
1566:
777:{\displaystyle x\mapsto f(x)}
129:occurs when an author uses a
302:{\displaystyle \mathbb {Z} }
7:
1549:
992:One example of this is the
257:
252:Cartesian coordinate system
10:
1686:
317:of integers together with
309:may denote the set of the
141:should be contrasted with
1427:, the equivalence class {
901:{\displaystyle x^{2}+x+1}
858:{\displaystyle x^{2}+x+1}
819:{\displaystyle x^{2}+x+1}
285:suppression of parameters
1670:Mathematical terminology
977:Equality vs. isomorphism
147:constants of integration
1304:{\displaystyle z=(y,z)}
1266:{\displaystyle (x,y)=x}
197:, it is common to call
1373:
1305:
1267:
1229:
1155:
1154:{\displaystyle z\in G}
1129:
1128:{\displaystyle y\in F}
1103:
1102:{\displaystyle x\in E}
1073:
967:
922:
902:
859:
820:
778:
743:
723:
703:
683:
663:
634:
607:
570:
529:
493:
437:
403:the ring of integers.
397:
357:
303:
277:mathematical operation
240:
207:canonically isomorphic
170:properly designates a
1665:Mathematical notation
1556:Mathematical notation
1383:—using the idea of a
1374:
1306:
1268:
1230:
1156:
1130:
1104:
1074:
968:
923:
903:
860:
821:
779:
744:
724:
704:
684:
669:denotes the value of
664:
635:
608:
571:
530:
494:
438:
398:
358:
304:
241:
201:"a representation of
158:abuse of terminology,
152:A related concept is
131:mathematical notation
1639:www.abstractmath.org
1612:www.abstractmath.org
1405:equivalence relation
1315:
1277:
1239:
1165:
1139:
1113:
1087:
1003:
932:
912:
873:
830:
791:
753:
733:
713:
693:
673:
662:{\displaystyle f(x)}
644:
621:
606:{\displaystyle f(x)}
588:
539:
503:
479:
420:
367:
333:
291:
269:mathematical objects
221:
43:improve this article
1585:math.vanderbilt.edu
1494:Lebesgue integrable
1395:Equivalence classes
1385:natural isomorphism
826:...", when in fact
58:"Abuse of notation"
1536:category theoretic
1448:modular arithmetic
1369:
1301:
1263:
1225:
1151:
1125:
1099:
1069:
983:universal property
963:
918:
898:
855:
816:
774:
739:
719:
699:
679:
659:
633:{\displaystyle f,}
630:
603:
566:
525:
489:
433:
410:consists of a set
393:
353:
299:
236:
172:group homomorphism
1542:can be seen as a
1498:almost everywhere
1401:equivalence class
1235:would imply that
994:Cartesian product
921:{\displaystyle x}
742:{\displaystyle x}
722:{\displaystyle f}
702:{\displaystyle x}
682:{\displaystyle f}
580:Function notation
475:and its topology
408:topological space
211:constant function
154:abuse of language
139:Abuse of notation
127:abuse of notation
119:
118:
111:
93:
1677:
1649:
1648:
1646:
1645:
1631:
1622:
1621:
1619:
1618:
1604:
1595:
1594:
1592:
1591:
1577:
1541:
1533:
1529:
1524:partial function
1521:
1492:, or classes of
1446:For example, in
1399:Referring to an
1378:
1376:
1375:
1370:
1310:
1308:
1307:
1302:
1272:
1270:
1269:
1264:
1234:
1232:
1231:
1226:
1160:
1158:
1157:
1152:
1134:
1132:
1131:
1126:
1108:
1106:
1105:
1100:
1078:
1076:
1075:
1070:
972:
970:
969:
964:
950:
949:
927:
925:
924:
919:
907:
905:
904:
899:
885:
884:
864:
862:
861:
856:
842:
841:
825:
823:
822:
817:
803:
802:
783:
781:
780:
775:
748:
746:
745:
740:
728:
726:
725:
720:
708:
706:
705:
700:
689:for the element
688:
686:
685:
680:
668:
666:
665:
660:
639:
637:
636:
631:
612:
610:
609:
604:
575:
573:
572:
567:
562:
558:
557:
534:
532:
531:
526:
521:
520:
498:
496:
495:
490:
488:
487:
474:
468:
462:
452:
442:
440:
439:
434:
429:
428:
415:
402:
400:
399:
394:
377:
362:
360:
359:
354:
343:
308:
306:
305:
300:
298:
287:). For example,
245:
243:
242:
237:
235:
234:
229:
215:binary operation
114:
107:
103:
100:
94:
92:
51:
27:
19:
1685:
1684:
1680:
1679:
1678:
1676:
1675:
1674:
1655:
1654:
1653:
1652:
1643:
1641:
1633:
1632:
1625:
1616:
1614:
1606:
1605:
1598:
1589:
1587:
1579:
1578:
1574:
1569:
1552:
1539:
1538:context, where
1531:
1527:
1509:
1506:
1469:if and only if
1397:
1381:category theory
1316:
1313:
1312:
1278:
1275:
1274:
1240:
1237:
1236:
1166:
1163:
1162:
1161:, the identity
1140:
1137:
1136:
1114:
1111:
1110:
1088:
1085:
1084:
1004:
1001:
1000:
979:
945:
941:
933:
930:
929:
928:can be denoted
913:
910:
909:
880:
876:
874:
871:
870:
837:
833:
831:
828:
827:
798:
794:
792:
789:
788:
754:
751:
750:
734:
731:
730:
714:
711:
710:
694:
691:
690:
674:
671:
670:
645:
642:
641:
622:
619:
618:
589:
586:
585:
582:
553:
552:
551:
540:
537:
536:
516:
515:
504:
501:
500:
483:
482:
480:
477:
476:
470:
464:
458:
448:
424:
423:
421:
418:
417:
411:
373:
368:
365:
364:
339:
334:
331:
330:
294:
292:
289:
288:
265:
260:
248:Euclidean space
230:
225:
224:
222:
219:
218:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
1683:
1673:
1672:
1667:
1651:
1650:
1623:
1596:
1571:
1570:
1568:
1565:
1564:
1563:
1558:
1551:
1548:
1505:
1502:
1396:
1393:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1150:
1147:
1144:
1124:
1121:
1118:
1098:
1095:
1092:
1081:
1080:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
978:
975:
962:
959:
956:
953:
948:
944:
940:
937:
917:
897:
894:
891:
888:
883:
879:
854:
851:
848:
845:
840:
836:
815:
812:
809:
806:
801:
797:
773:
770:
767:
764:
761:
758:
738:
718:
698:
678:
658:
655:
652:
649:
629:
626:
602:
599:
596:
593:
581:
578:
565:
561:
556:
550:
547:
544:
524:
519:
514:
511:
508:
486:
432:
427:
392:
389:
386:
383:
380:
376:
372:
352:
349:
346:
342:
338:
327:multiplication
297:
264:
261:
259:
256:
233:
228:
167:representation
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
1682:
1671:
1668:
1666:
1663:
1662:
1660:
1640:
1636:
1630:
1628:
1613:
1609:
1603:
1601:
1586:
1582:
1576:
1572:
1562:
1559:
1557:
1554:
1553:
1547:
1545:
1537:
1525:
1520:
1516:
1512:
1501:
1499:
1495:
1491:
1490:measure space
1486:
1484:
1480:
1476:
1472:
1468:
1464:
1460:
1457:
1453:
1449:
1444:
1442:
1438:
1434:
1430:
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1392:
1388:
1386:
1382:
1363:
1360:
1357:
1354:
1351:
1345:
1339:
1336:
1330:
1327:
1324:
1295:
1292:
1289:
1283:
1280:
1260:
1257:
1251:
1248:
1245:
1216:
1213:
1210:
1204:
1201:
1195:
1189:
1186:
1180:
1177:
1174:
1148:
1145:
1142:
1122:
1119:
1116:
1096:
1093:
1090:
1066:
1063:
1060:
1057:
1054:
1051:
1045:
1042:
1039:
1033:
1030:
1027:
1024:
1021:
1015:
1012:
1009:
999:
998:
997:
995:
990:
988:
984:
974:
960:
957:
954:
951:
946:
942:
935:
915:
895:
892:
889:
886:
881:
877:
868:
852:
849:
846:
843:
838:
834:
813:
810:
807:
804:
799:
795:
785:
768:
762:
756:
749:..." or "Let
736:
716:
696:
676:
653:
647:
627:
624:
616:
597:
591:
577:
559:
548:
545:
512:
509:
473:
467:
461:
456:
451:
446:
430:
414:
409:
406:Similarly, a
404:
387:
384:
381:
378:
347:
344:
328:
324:
320:
316:
312:
286:
282:
278:
274:
271:consist of a
270:
255:
253:
249:
231:
216:
212:
208:
204:
200:
196:
192:
188:
186:
180:
177:
173:
169:
168:
163:
159:
155:
150:
148:
144:
140:
136:
132:
128:
124:
113:
110:
102:
99:December 2017
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
1642:. Retrieved
1638:
1615:. Retrieved
1611:
1588:. Retrieved
1584:
1575:
1518:
1514:
1510:
1507:
1504:Subjectivity
1487:
1482:
1478:
1474:
1470:
1466:
1462:
1458:
1452:finite group
1445:
1440:
1436:
1432:
1428:
1424:
1420:
1412:
1408:
1398:
1389:
1082:
991:
980:
786:
583:
471:
465:
459:
449:
412:
405:
284:
266:
202:
198:
195:vector space
190:
184:
178:
166:
161:
157:
153:
151:
142:
138:
126:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
1417:partitioned
123:mathematics
1659:Categories
1644:2019-11-03
1617:2019-11-03
1590:2019-11-03
1567:References
987:isomorphic
867:polynomial
69:newspapers
1311:, and so
1146:∈
1120:∈
1094:∈
1064:×
1058:×
1043:×
1034:×
1022:×
1013:×
939:↦
760:↦
455:open sets
388:⋅
321:, or the
135:intuition
1561:Misnomer
1550:See also
1544:morphism
1522:" for a
1513: :
1473:≡
615:function
560:′
319:addition
311:integers
281:topology
258:Examples
189:, where
160:where a
445:subsets
174:from a
83:scholar
1403:of an
313:, the
143:misuse
85:
78:
71:
64:
56:
1526:from
1485:− 1.
1477:(mod
1456:order
865:is a
453:(the
315:group
279:or a
267:Many
193:is a
176:group
90:JSTOR
76:books
1450:, a
1273:and
1135:and
640:and
535:and
323:ring
246:the
162:term
62:news
1530:to
1500:".
1454:of
1415:is
1407:by
908:to
617:is
447:of
273:set
183:GL(
181:to
156:or
149:).
121:In
45:by
1661::
1637:.
1626:^
1610:.
1599:^
1583:.
1517:→
1465:~
1439:~
1435:|
1431:∈
1423:∈
1387:.
1109:,
961:1.
254:.
125:,
1647:.
1620:.
1593:.
1540:f
1532:B
1528:A
1519:B
1515:A
1511:f
1483:n
1479:n
1475:y
1471:x
1467:y
1463:x
1459:n
1441:x
1437:y
1433:X
1429:y
1425:X
1421:x
1413:X
1409:x
1367:)
1364:z
1361:,
1358:y
1355:,
1352:x
1349:(
1346:=
1343:)
1340:z
1337:,
1334:)
1331:y
1328:,
1325:x
1322:(
1319:(
1299:)
1296:z
1293:,
1290:y
1287:(
1284:=
1281:z
1261:x
1258:=
1255:)
1252:y
1249:,
1246:x
1243:(
1223:)
1220:)
1217:z
1214:,
1211:y
1208:(
1205:,
1202:x
1199:(
1196:=
1193:)
1190:z
1187:,
1184:)
1181:y
1178:,
1175:x
1172:(
1169:(
1149:G
1143:z
1123:F
1117:y
1097:E
1091:x
1079:.
1067:G
1061:F
1055:E
1052:=
1049:)
1046:G
1040:F
1037:(
1031:E
1028:=
1025:G
1019:)
1016:F
1010:E
1007:(
958:+
955:x
952:+
947:2
943:x
936:x
916:x
896:1
893:+
890:x
887:+
882:2
878:x
853:1
850:+
847:x
844:+
839:2
835:x
814:1
811:+
808:x
805:+
800:2
796:x
772:)
769:x
766:(
763:f
757:x
737:x
717:f
697:x
677:f
657:)
654:x
651:(
648:f
628:,
625:f
601:)
598:x
595:(
592:f
564:)
555:T
549:,
546:X
543:(
523:)
518:T
513:,
510:X
507:(
485:T
472:X
466:X
460:X
450:X
431:,
426:T
413:X
391:)
385:,
382:+
379:,
375:Z
371:(
351:)
348:+
345:,
341:Z
337:(
296:Z
232:3
227:R
203:G
199:V
191:V
187:)
185:V
179:G
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
