638:+ 1. The base case is often trivial and is identified as such, although there are situations where the base case is difficult but the inductive step is trivial. Similarly, one might want to prove that some property is possessed by all the members of a certain set. The main part of the proof will consider the case of a nonempty set, and examine the members in detail; in the case where the set is empty, the property is trivially possessed by all the members of the empty set, since there are none (see
745:
845:
Two mathematicians who are discussing a theorem: the first mathematician says that the theorem is "trivial". In response to the other's request for an explanation, he then proceeds with twenty minutes of exposition. At the end of the explanation, the second mathematician agrees that the theorem is
645:
The judgement of whether a situation under consideration is trivial or not depends on who considers it since the situation is obviously true for someone who has sufficient knowledge or experience of it while to someone who has never seen this, it may be even hard to be understood so not trivial at
72:
The judgement of whether a situation under consideration is trivial or not depends on who considers it since the situation is obviously true for someone who has sufficient knowledge or experience of it while to someone who has never seen this, it may be even hard to be understood so not trivial at
646:
all. And there can be an argument about how quickly and easily a problem should be recognized for the problem to be treated as trivial. The following examples show the subjectivity and ambiguity of the triviality judgement.
73:
all. And there can be an argument about how quickly and easily a problem should be recognized for the problem to be treated as trivial. So, triviality is not a universally agreed property in mathematics and logic.
919:
394:
1113:
989:
1083:
842:
A common joke in the mathematical community is to say that "trivial" is synonymous with "proved"—that is, any theorem can be considered "trivial" once it is known to be proved as true.
1210:
has zeros at the negative even numbers −2, −4, … Though the proof is comparatively easy, this result would still not normally be called trivial; however, it is in this case, for its
695:, is always true. Here, the proof follows immediately by virtue of the definition of material implication in which as the implication is true regardless of the truth value of the
555:
1122:, there is a very simple group with just one element in it; this is often called the "trivial group". All other groups, which are more complicated, are called "nontrivial".
1053:
1031:
444:
337:
171:
597:
241:
1189:
1163:
487:
281:
208:
849:
When a mathematician says that a theorem is trivial, but he is unable to prove it by himself at the moment that he pronounces it as trivial, is the theorem trivial?
81:
In mathematics, the term "trivial" is often used to refer to objects (e.g., groups, topological spaces) with a very simple structure. These include, among others:
1009:
762:
678:
657:, the proof may very well hinge on the remark that any natural number has a successor – a statement which should itself be proved or be taken as an
653:
would probably, given a number, trivially assume the existence of a larger number. However, when proving basic results about the natural numbers in
57:
usually refers to a simple technical aspect of some proof or definition. The origin of the term in mathematical language comes from the medieval
1218:). Accordingly, the negative even numbers are called the trivial zeros of the function, while any other zeros are considered to be non-trivial.
45:
is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g.,
1431:
859:
69:, which is commonly used to indicate that an example or a solution is not simple, or that a statement or a theorem is not easy to prove.
809:
1519:
1485:
1458:
1384:
1349:
828:
781:
788:
766:
1203:, so this type of dependence is called "trivial". All other dependences, which are less obvious, are called "nontrivial".
1233:
795:
489:, which is considered obvious and hence is called the "trivial" solution. In some cases, there may be other solutions (
1411:
1238:
133:
that have a very simple structure, but for the sake of completeness cannot be omitted. These solutions are called the
1243:
1088:
777:
755:
297:
961:
1228:
1058:
654:
497:
622:
have two parts: the "base case" which shows that the theorem is true for a particular initial value (such as
251:
846:
trivial. But can we say that this theorem is trivial even if it takes a lot of time and effort to prove it?
1258:
619:
344:
211:
145:
852:
Often, as a joke, a problem is referred to as "intuitively obvious". For example, someone experienced in
1207:
1136:
734:
138:
721:
is false. In this case, the implication is always true regardless of the truth value of the consequent
507:
921:
However, to someone with no knowledge of integral calculus, this is not obvious, so it is not trivial.
955:
290:
1214:
zeros are generally unknown and have important applications and involve open questions (such as the
802:
951:. These are called "trivial factors". Any other factor, if it exists, would be called "nontrivial".
650:
615:
99:
58:
46:
1036:
1014:
1215:
696:
399:
112:
1481:
1475:
1454:
1448:
1427:
1423:
1380:
1374:
1355:
1345:
662:
447:
89:
50:
564:
1415:
1263:
1168:
1142:
630:= 1), and the inductive step which shows that if the theorem is true for a certain value of
457:
178:
103:
1253:
1132:
618:
of a proof, which for the sake of completeness cannot be ignored. For instance, proofs by
603:, but such solutions are obvious and obtainable with little effort, and hence "trivial".
221:
1407:
994:
1513:
1416:
932:
706:
639:
451:
116:
95:
1126:
1119:
561:
is greater than 2. Clearly, there are some solutions to the equation. For example,
108:
744:
38:
1248:
689:
244:
215:
62:
28:
1504:
1359:
1289:
490:
85:
446:
is important in mathematics and physics, as it could be used to describe a
1316:
17:
853:
130:
936:
1196:
1129:, the trivial graph is a graph which has only 1 vertex and no edge.
1085:. This is called the "trivial solution". Any other solutions, with
658:
1376:
Introduction to
Partial Differential Equations with Applications
738:
725:– again by virtue of the definition of material implication.
1453:(2nd, illustrated ed.). Berlin: Springer. p. 250.
1418:
Mathematical proofs: a transition to advanced mathematics
1422:(2nd ed.). Boston: Pearson/Addison Wesley. p.
61:
curriculum, which distinguishes from the more difficult
914:{\displaystyle \int _{0}^{1}x^{2}\,dx={\frac {1}{3}}.}
1171:
1145:
1091:
1061:
1039:
1017:
997:
964:
862:
567:
510:
460:
402:
347:
300:
254:
224:
181:
148:
769:. Unsourced material may be challenged and removed.
1183:
1157:
1107:
1077:
1047:
1025:
1003:
983:
913:
591:
549:
481:
438:
388:
331:
275:
235:
202:
165:
856:would consider the following statement trivial:
649:Triviality also depends on context. A proof in
1108:{\displaystyle \mathbf {x} \neq \mathbf {0} }
454:on a string. It always includes the solution
129:can also be used to describe solutions to an
8:
1055:is the zero vector, has an obvious solution
493:), which are called "nontrivial" solutions.
1402:
1400:
1398:
1396:
1373:Zachmanoglou, E. C.; Thoe, Dale W. (1986).
1480:(Sixth ed.). CRC Press. p. 502.
1344:. University of Texas Press. p. 542.
984:{\displaystyle A\mathbf {x} =\mathbf {0} }
1170:
1144:
1100:
1092:
1090:
1078:{\displaystyle \mathbf {x} =\mathbf {0} }
1070:
1062:
1060:
1040:
1038:
1018:
1016:
996:
976:
968:
963:
898:
888:
882:
872:
867:
861:
829:Learn how and when to remove this message
566:
541:
528:
515:
509:
496:Similarly, mathematicians often describe
459:
401:
346:
320:
299:
253:
223:
180:
147:
1477:Mathematics for Engineers and Scientists
1275:
65:curriculum. The opposite of trivial is
634:, then it is also true for the value
7:
1311:
1309:
1283:
1281:
1279:
767:adding citations to reliable sources
702:if the consequent is fixed as true.
389:{\displaystyle f''(x)=-\lambda f(x)}
947:has four obvious factors: ±1 and ±
677:refers to a statement involving a
504:integer solutions to the equation
25:
661:so is not trivial (for more, see
550:{\displaystyle a^{n}+b^{n}=c^{n}}
1101:
1093:
1071:
1063:
1041:
1019:
977:
969:
935:, it is often important to find
743:
77:Trivial and nontrivial solutions
754:needs additional citations for
500:as asserting that there are no
1175:
1149:
470:
464:
427:
421:
412:
406:
383:
377:
362:
356:
310:
304:
264:
258:
243:. The trivial solution is the
197:
191:
1:
778:"Triviality" mathematics
92:containing no or null members
1234:Initial and terminal objects
1048:{\displaystyle \mathbf {0} }
1026:{\displaystyle \mathbf {x} }
137:. For example, consider the
1450:Number Theory for Computing
1239:List of mathematical jargon
614:may also refer to any easy
450:in quantum mechanics, or a
439:{\displaystyle f(0)=f(L)=0}
332:{\displaystyle y(x)=e^{x}.}
1536:
1505:Trivial entry at MathWorld
1342:Dictionary of word origins
1115:, are called "nontrivial".
1033:is an unknown vector, and
732:
713:in a material implication
341:The differential equation
26:
607:In mathematical reasoning
396:with boundary conditions
1520:Mathematical terminology
655:elementary number theory
27:Not to be confused with
1410:; Polimeni, Albert D.;
1208:Riemann's zeta function
709:, where the antecedent
705:A related concept is a
592:{\displaystyle a=b=c=0}
1474:Jeffrey, Alan (2004).
1259:Trivial representation
1185:
1184:{\displaystyle X\to Y}
1159:
1158:{\displaystyle X\to Y}
1109:
1079:
1049:
1027:
1005:
985:
915:
620:mathematical induction
599:is a solution for any
593:
551:
483:
482:{\displaystyle f(x)=0}
440:
390:
333:
277:
276:{\displaystyle y(x)=0}
237:
204:
203:{\displaystyle y=y(x)}
167:
34:Mathematically obvious
1447:Yan, Song Y. (2002).
1294:mathworld.wolfram.com
1206:It can be shown that
1186:
1160:
1137:functional dependency
1135:has a concept called
1110:
1080:
1050:
1028:
1006:
986:
939:of an integer number
916:
735:Proof by intimidation
733:Further information:
594:
552:
498:Fermat's last theorem
484:
441:
391:
334:
278:
238:
205:
168:
139:differential equation
1317:"Mathwords: Trivial"
1169:
1143:
1089:
1059:
1037:
1015:
995:
962:
860:
763:improve this article
679:material implication
565:
508:
458:
400:
345:
298:
291:exponential function
252:
222:
179:
166:{\displaystyle y'=y}
146:
102:containing only the
1340:Ayto, John (1990).
1288:Weisstein, Eric W.
1011:is a fixed matrix,
877:
651:functional analysis
98:: the mathematical
1216:Riemann hypothesis
1181:
1155:
1105:
1075:
1045:
1023:
1001:
981:
911:
863:
589:
547:
479:
436:
386:
329:
273:
236:{\displaystyle y'}
233:
200:
163:
51:topological spaces
1433:978-0-3-2139053-0
1321:www.mathwords.com
1165:. The dependence
1004:{\displaystyle A}
906:
839:
838:
831:
813:
673:In some texts, a
448:particle in a box
135:trivial solutions
16:(Redirected from
1527:
1492:
1491:
1471:
1465:
1464:
1444:
1438:
1437:
1421:
1404:
1391:
1390:
1370:
1364:
1363:
1337:
1331:
1330:
1328:
1327:
1313:
1304:
1303:
1301:
1300:
1285:
1264:Trivial topology
1190:
1188:
1187:
1182:
1164:
1162:
1161:
1156:
1114:
1112:
1111:
1106:
1104:
1096:
1084:
1082:
1081:
1076:
1074:
1066:
1054:
1052:
1051:
1046:
1044:
1032:
1030:
1029:
1024:
1022:
1010:
1008:
1007:
1002:
990:
988:
987:
982:
980:
972:
954:The homogeneous
920:
918:
917:
912:
907:
899:
887:
886:
876:
871:
834:
827:
823:
820:
814:
812:
771:
747:
739:
598:
596:
595:
590:
556:
554:
553:
548:
546:
545:
533:
532:
520:
519:
488:
486:
485:
480:
445:
443:
442:
437:
395:
393:
392:
387:
355:
338:
336:
335:
330:
325:
324:
289:solution is the
282:
280:
279:
274:
242:
240:
239:
234:
232:
209:
207:
206:
201:
172:
170:
169:
164:
156:
104:identity element
41:, the adjective
21:
1535:
1534:
1530:
1529:
1528:
1526:
1525:
1524:
1510:
1509:
1501:
1496:
1495:
1488:
1473:
1472:
1468:
1461:
1446:
1445:
1441:
1434:
1408:Chartrand, Gary
1406:
1405:
1394:
1387:
1379:. p. 309.
1372:
1371:
1367:
1352:
1339:
1338:
1334:
1325:
1323:
1315:
1314:
1307:
1298:
1296:
1287:
1286:
1277:
1272:
1254:Trivial measure
1225:
1167:
1166:
1141:
1140:
1133:Database theory
1087:
1086:
1057:
1056:
1035:
1034:
1013:
1012:
993:
992:
960:
959:
928:
878:
858:
857:
835:
824:
818:
815:
772:
770:
760:
748:
737:
731:
671:
609:
563:
562:
537:
524:
511:
506:
505:
456:
455:
398:
397:
348:
343:
342:
316:
296:
295:
250:
249:
225:
220:
219:
177:
176:
149:
144:
143:
79:
35:
32:
23:
22:
15:
12:
11:
5:
1533:
1531:
1523:
1522:
1512:
1511:
1508:
1507:
1500:
1499:External links
1497:
1494:
1493:
1486:
1466:
1459:
1439:
1432:
1392:
1385:
1365:
1350:
1332:
1305:
1274:
1273:
1271:
1268:
1267:
1266:
1261:
1256:
1251:
1246:
1241:
1236:
1231:
1224:
1221:
1220:
1219:
1204:
1180:
1177:
1174:
1154:
1151:
1148:
1130:
1123:
1116:
1103:
1099:
1095:
1073:
1069:
1065:
1043:
1021:
1000:
979:
975:
971:
967:
952:
927:
924:
923:
922:
910:
905:
902:
897:
894:
891:
885:
881:
875:
870:
866:
850:
847:
843:
837:
836:
751:
749:
742:
730:
727:
670:
669:Trivial proofs
667:
663:Peano's axioms
608:
605:
588:
585:
582:
579:
576:
573:
570:
544:
540:
536:
531:
527:
523:
518:
514:
478:
475:
472:
469:
466:
463:
435:
432:
429:
426:
423:
420:
417:
414:
411:
408:
405:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
354:
351:
328:
323:
319:
315:
312:
309:
306:
303:
272:
269:
266:
263:
260:
257:
231:
228:
199:
196:
193:
190:
187:
184:
162:
159:
155:
152:
120:
119:
106:
93:
78:
75:
33:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1532:
1521:
1518:
1517:
1515:
1506:
1503:
1502:
1498:
1489:
1487:1-58488-488-6
1483:
1479:
1478:
1470:
1467:
1462:
1460:3-540-43072-5
1456:
1452:
1451:
1443:
1440:
1435:
1429:
1425:
1420:
1419:
1413:
1409:
1403:
1401:
1399:
1397:
1393:
1388:
1386:9780486652511
1382:
1378:
1377:
1369:
1366:
1361:
1357:
1353:
1351:1-55970-214-1
1347:
1343:
1336:
1333:
1322:
1318:
1312:
1310:
1306:
1295:
1291:
1284:
1282:
1280:
1276:
1269:
1265:
1262:
1260:
1257:
1255:
1252:
1250:
1247:
1245:
1242:
1240:
1237:
1235:
1232:
1230:
1227:
1226:
1222:
1217:
1213:
1209:
1205:
1202:
1198:
1194:
1178:
1172:
1152:
1146:
1138:
1134:
1131:
1128:
1124:
1121:
1117:
1097:
1067:
998:
973:
965:
957:
953:
950:
946:
943:. Any number
942:
938:
934:
933:number theory
930:
929:
925:
908:
903:
900:
895:
892:
889:
883:
879:
873:
868:
864:
855:
851:
848:
844:
841:
840:
833:
830:
822:
811:
808:
804:
801:
797:
794:
790:
787:
783:
780: –
779:
775:
774:Find sources:
768:
764:
758:
757:
752:This article
750:
746:
741:
740:
736:
728:
726:
724:
720:
716:
712:
708:
707:vacuous truth
703:
701:
698:
694:
691:
687:
683:
680:
676:
675:trivial proof
668:
666:
664:
660:
656:
652:
647:
643:
641:
640:vacuous truth
637:
633:
629:
625:
621:
617:
613:
606:
604:
602:
586:
583:
580:
577:
574:
571:
568:
560:
542:
538:
534:
529:
525:
521:
516:
512:
503:
499:
494:
492:
476:
473:
467:
461:
453:
452:standing wave
449:
433:
430:
424:
418:
415:
409:
403:
380:
374:
371:
368:
365:
359:
352:
349:
339:
326:
321:
317:
313:
307:
301:
293:
292:
288:
283:
270:
267:
261:
255:
247:
246:
245:zero function
229:
226:
217:
213:
194:
188:
185:
182:
173:
160:
157:
153:
150:
141:
140:
136:
132:
128:
124:
118:
117:singleton set
115:defined on a
114:
110:
107:
105:
101:
97:
96:Trivial group
94:
91:
87:
84:
83:
82:
76:
74:
70:
68:
64:
60:
56:
52:
48:
44:
40:
30:
19:
1476:
1469:
1449:
1442:
1417:
1375:
1368:
1341:
1335:
1324:. Retrieved
1320:
1297:. Retrieved
1293:
1244:Pathological
1211:
1200:
1192:
1127:graph theory
1120:group theory
948:
944:
940:
825:
819:January 2023
816:
806:
799:
792:
785:
773:
761:Please help
756:verification
753:
722:
718:
714:
710:
704:
699:
692:
685:
681:
674:
672:
648:
644:
635:
631:
627:
623:
611:
610:
600:
558:
501:
495:
340:
294:
286:
284:
248:
174:
142:
134:
126:
122:
121:
109:Trivial ring
80:
71:
66:
54:
53:). The noun
42:
36:
1412:Zhang, Ping
1191:is true if
642:for more).
39:mathematics
1326:2019-12-14
1299:2019-12-14
1270:References
1249:Trivialism
1229:Degeneracy
1139:, written
789:newspapers
697:antecedent
690:consequent
688:where the
502:nontrivial
287:nontrivial
216:derivative
67:nontrivial
63:quadrivium
55:triviality
29:Trivialism
18:Nontrivial
1290:"Trivial"
1176:→
1150:→
1098:≠
958:equation
865:∫
491:sinusoids
372:λ
369:−
86:Empty set
1514:Category
1414:(2008).
1360:33022699
1223:See also
991:, where
926:Examples
854:calculus
557:, where
353:″
285:while a
230:′
212:function
154:′
131:equation
937:factors
803:scholar
626:= 0 or
612:Trivial
125:Trivial
59:trivium
43:trivial
1484:
1457:
1430:
1383:
1358:
1348:
1197:subset
956:matrix
805:
798:
791:
784:
776:
214:whose
175:where
88:: the
47:groups
1212:other
1195:is a
810:JSTOR
796:books
729:Humor
659:axiom
210:is a
100:group
1482:ISBN
1455:ISBN
1428:ISBN
1381:ISBN
1356:OCLC
1346:ISBN
782:news
616:case
113:ring
111:: a
1199:of
1125:In
1118:In
931:In
765:by
665:).
218:is
90:set
37:In
1516::
1426:.
1424:68
1395:^
1354:.
1319:.
1308:^
1292:.
1278:^
686:Q,
49:,
1490:.
1463:.
1436:.
1389:.
1362:.
1329:.
1302:.
1201:X
1193:Y
1179:Y
1173:X
1153:Y
1147:X
1102:0
1094:x
1072:0
1068:=
1064:x
1042:0
1020:x
999:A
978:0
974:=
970:x
966:A
949:N
945:N
941:N
909:.
904:3
901:1
896:=
893:x
890:d
884:2
880:x
874:1
869:0
832:)
826:(
821:)
817:(
807:·
800:·
793:·
786:·
759:.
723:Q
719:Q
717:→
715:P
711:P
700:P
693:Q
684:→
682:P
636:n
632:n
628:n
624:n
601:n
587:0
584:=
581:c
578:=
575:b
572:=
569:a
559:n
543:n
539:c
535:=
530:n
526:b
522:+
517:n
513:a
477:0
474:=
471:)
468:x
465:(
462:f
434:0
431:=
428:)
425:L
422:(
419:f
416:=
413:)
410:0
407:(
404:f
384:)
381:x
378:(
375:f
366:=
363:)
360:x
357:(
350:f
327:.
322:x
318:e
314:=
311:)
308:x
305:(
302:y
271:0
268:=
265:)
262:x
259:(
256:y
227:y
198:)
195:x
192:(
189:y
186:=
183:y
161:y
158:=
151:y
127:"
123:"
31:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.