43:
935:
Book on the history of the mathematical theory of Fuzzy Sets: The
Fuzzification of Systems. The Genesis of Fuzzy Set Theory and Its Initial Applications -- Developments up to the 1970s (Studies in Fuzziness and Soft Computing, Vol. 216) Berlin, New York, : Springer
568:
Analogues of other mathematical subjects have been translated to fuzzy mathematics, such as fuzzy field theory and fuzzy Galois theory, fuzzy topology, fuzzy geometry, fuzzy orderings, and fuzzy graphs.
107:
that deals with partial inclusion of elements in a set on a spectrum, as opposed to simple binary "yes" or "no" (0 or 1) inclusion. It started in 1965 after the publication of
61:
849:
Yeh, R.T., Bang, S.Y. (1975) "Fuzzy graphs, fuzzy relations and their applications to cluster analysis". In: Zadeh, L.A., Fu, K.S., Tanaka, K., Shimura, M. (eds.),
217:
Usually, a fuzzification of mathematical concepts is based on a generalization of these concepts from characteristic functions to membership functions. Let
645:
858:
837:
933:
889:
79:
922:
350:
An important generalization principle used in fuzzification of algebraic operations is a closure property. Let * be a
31:
347:
operations because in this case more properties of traditional mathematics can be extended to the fuzzy case.
161:
142:
588:
164:. This function is also called a membership function. A membership function is a generalization of an
949:
578:
895:
828:
A. Rosenfeld, A. (1975) "Fuzzy graphs". In: Zadeh, L.A., Fu, K.S., Tanaka, K., Shimura, M. (eds.),
478:
411:
195:
The evolution of the fuzzification of mathematical concepts can be broken down into three stages:
802:
593:
169:
165:
854:
833:
135:
108:
583:
351:
177:
100:
743:. Advances in Fuzzy Systems - Applications and Theory, vol. 9, World Scientific, Singapore.
926:
905:
204:
the explosion of the possible choices in the generalization process during the eighties,
943:
915:
909:
899:
879:
736:
875:
477:
A similar generalization principle is used, for example, for fuzzification of the
885:
643:
Kerre, E.E., Mordeson, J.N. (2005) "A historical overview of fuzzy mathematics",
17:
565:
Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld.
116:
104:
96:
761:
Buckley, J.J., Eslami, E. (1997) "Fuzzy plane geometry I: Points and lines".
919:
787:
Chakraborty, D. and Ghosh, D. (2014) "Analytical fuzzy plane geometry II".
774:
Ghosh, D., Chakraborty, D. (2012) "Analytical fuzzy plane geometry I".
851:
Fuzzy Sets and their
Applications to Cognitive and Decision Processes
830:
Fuzzy Sets and their
Applications to Cognitive and Decision Processes
701:. Studies in Fuzziness and Soft Computing, vol. 182. Springer-Verlag.
598:
324:
688:. Studies in Fuzziness and Soft Computing, vol. 131, Springer-Verlag
201:
straightforward fuzzification during the sixties and seventies,
800:
Zadeh L.A. (1971) "Similarity relations and fuzzy orderings".
36:
119:
is an example of a field that utilizes fuzzy set theory.
57:
339:. A straightforward fuzzification is usually based on
697:
Mordeson, J.N., Bhutani, K.R., Rosenfeld, A. (2005)
52:
may be too technical for most readers to understand
817:Introduction a la théorie des sous-ensembles flows
176:= {0, 1}. More generally, one can use any
723:Chang, C.L. (1968) "Fuzzy topological spaces",
684:Mordeson, J.N., Malik, D.S., Kuroli, N. (2003)
327:and t-conorm, respectively; for example, min(
8:
658:Klement, E.P., Mesiar, R., Pap, E. (2000)
358:. The closure property for a fuzzy subset
929:- Web site hosted at Creighton University
207:the standardization, axiomatization, and
80:Learn how and when to remove this message
64:, without removing the technical details.
646:New Mathematics and Natural Computation
610:
671:Rosenfeld, A. (1971) "Fuzzy groups",
62:make it understandable to non-experts
7:
335:) can be replaced by multiplication
890:Stanford Encyclopedia of Philosophy
710:Mordeson, J.N., Malik, D.S (1998)
630:Goguen, J. (1967) "L-fuzzy sets",
617:Zadeh, L. A. (1965) "Fuzzy sets",
183:in a definition of a fuzzy subset
25:
505:is (fuzzy-)transitive if for all
41:
211:-fuzzification in the nineties.
752:Poston, Tim, "Fuzzy Geometry".
1:
896:Triangular Norms and Conorms
853:, Academic Press, New York,
832:, Academic Press, New York,
918:Mathematics of Uncertainty
32:Fuzzy math (disambiguation)
966:
172:) of a subset defined for
29:
712:Fuzzy Commutative Algebra
251:are defined as follows: (
225:be two fuzzy subsets of
619:Information and Control
485:be a fuzzy relation on
170:characteristic function
789:Fuzzy Sets and Systems
776:Fuzzy Sets and Systems
763:Fuzzy Sets and Systems
904:Dubois, D., Prade H.
815:Kaufmann, A. (1973).
589:Monoidal t-norm logic
493:is a fuzzy subset of
27:Branch of mathematics
739:, Luo, M.-K. (1997)
725:J. Math. Anal. Appl.
673:J. Math. Anal. Appl.
662:. Dordrecht, Kluwer.
632:J. Math. Anal. Appl.
579:Fuzzy measure theory
30:For other uses, see
920:Fuzzy Math Research
714:. World Scientific.
925:2009-06-29 at the
906:Possibility Theory
699:Fuzzy Group Theory
594:Possibility theory
418:a fuzzy subset of
166:indicator function
859:978-0-12-775260-0
838:978-0-12-775260-0
109:Lotfi Asker Zadeh
95:is the branch of
93:Fuzzy mathematics
90:
89:
82:
18:Fuzzy Mathematics
16:(Redirected from
957:
862:
847:
841:
826:
820:
819:. Paris. Masson.
813:
807:
798:
792:
785:
779:
772:
766:
759:
753:
750:
744:
734:
728:
721:
715:
708:
702:
695:
689:
686:Fuzzy Semigroups
682:
676:
669:
663:
660:Triangular Norms
656:
650:
641:
635:
628:
622:
615:
584:Fuzzy subalgebra
366:is that for all
352:binary operation
346:
342:
322:
318:
178:complete lattice
111:'s seminal work
101:fuzzy set theory
85:
78:
74:
71:
65:
45:
44:
37:
21:
965:
964:
960:
959:
958:
956:
955:
954:
940:
939:
927:Wayback Machine
871:
866:
865:
848:
844:
827:
823:
814:
810:
799:
795:
786:
782:
773:
769:
760:
756:
751:
747:
735:
731:
722:
718:
709:
705:
696:
692:
683:
679:
670:
666:
657:
653:
642:
638:
629:
625:
616:
612:
607:
575:
563:
561:Fuzzy analogues
497: ×
344:
340:
320:
316:
283: ∪
193:
168:(also called a
125:
86:
75:
69:
66:
58:help improve it
55:
46:
42:
35:
28:
23:
22:
15:
12:
11:
5:
963:
961:
953:
952:
942:
941:
938:
937:
930:
912:
902:
892:
882:
870:
869:External links
867:
864:
863:
861:, pp. 125–149.
842:
821:
808:
793:
791:, 243, 84–109.
780:
767:
765:, 86, 179-187.
754:
745:
741:Fuzzy Topology
729:
727:, 24, 182—190.
716:
703:
690:
677:
675:, 35, 512-517.
664:
651:
636:
634:, 18, 145-174.
623:
609:
608:
606:
603:
602:
601:
596:
591:
586:
581:
574:
571:
562:
559:
481:property. Let
428:fuzzy subgroup
215:
214:
213:
212:
205:
202:
192:
189:
124:
121:
88:
87:
70:September 2015
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
962:
951:
948:
947:
945:
934:
931:
928:
924:
921:
917:
913:
911:
908:- article at
907:
903:
901:
898:- article at
897:
893:
891:
888:- article at
887:
883:
881:
878:- article at
877:
873:
872:
868:
860:
856:
852:
846:
843:
839:
835:
831:
825:
822:
818:
812:
809:
806:, 3, 177–200.
805:
804:
797:
794:
790:
784:
781:
778:, 209, 66-83.
777:
771:
768:
764:
758:
755:
749:
746:
742:
738:
733:
730:
726:
720:
717:
713:
707:
704:
700:
694:
691:
687:
681:
678:
674:
668:
665:
661:
655:
652:
648:
647:
640:
637:
633:
627:
624:
621:, 8, 338–353.
620:
614:
611:
604:
600:
597:
595:
592:
590:
587:
585:
582:
580:
577:
576:
572:
570:
566:
560:
558:
556:
552:
548:
544:
540:
536:
532:
528:
524:
520:
516:
512:
508:
504:
500:
496:
492:
488:
484:
480:
475:
473:
469:
465:
461:
457:
453:
449:
445:
441:
437:
433:
429:
425:
421:
417:
413:
409:
405:
401:
397:
393:
389:
385:
381:
377:
373:
369:
365:
361:
357:
353:
348:
338:
334:
330:
326:
315:. Instead of
314:
310:
306:
302:
298:
294:
290:
286:
282:
278:
274:
270:
266:
262:
258:
255: ∩
254:
250:
247: ∪
246:
243:
239:
236: ∩
235:
232:
228:
224:
220:
210:
206:
203:
200:
199:
198:
197:
196:
191:Fuzzification
190:
188:
186:
182:
179:
175:
171:
167:
163:
159:
155:
151:
147:
144:
140:
137:
133:
130:
122:
120:
118:
114:
110:
106:
102:
98:
94:
84:
81:
73:
63:
59:
53:
50:This article
48:
39:
38:
33:
19:
932:Seising, R.
910:Scholarpedia
900:Scholarpedia
880:Scholarpedia
874:Zadeh, L.A.
850:
845:
840:, pp. 77–95.
829:
824:
816:
811:
803:Inform. Sci.
801:
796:
788:
783:
775:
770:
762:
757:
748:
740:
732:
724:
719:
711:
706:
698:
693:
685:
680:
672:
667:
659:
654:
644:
639:
631:
626:
618:
613:
567:
564:
554:
550:
546:
542:
538:
534:
530:
526:
522:
518:
514:
510:
506:
502:
498:
494:
490:
486:
482:
479:transitivity
476:
471:
467:
463:
459:
455:
451:
447:
443:
439:
435:
431:
427:
423:
419:
415:
407:
403:
399:
395:
391:
387:
383:
379:
375:
371:
367:
363:
359:
355:
349:
336:
332:
328:
323:one can use
312:
308:
304:
300:
296:
292:
288:
284:
280:
276:
272:
268:
264:
260:
256:
252:
248:
244:
241:
237:
233:
231:intersection
230:
226:
222:
218:
216:
208:
194:
184:
180:
173:
157:
153:
149:
145:
138:
131:
129:fuzzy subset
128:
126:
112:
92:
91:
76:
67:
51:
950:Fuzzy logic
894:Navara, M.
886:Fuzzy Logic
876:Fuzzy Logic
434:if for all
307:)) for all
117:Linguistics
105:fuzzy logic
97:mathematics
884:Hajek, P.
737:Liu, Y.-M.
649:, 1, 1-26.
605:References
410:, *) be a
123:Definition
113:Fuzzy sets
99:including
406:)). Let (
944:Category
923:Archived
573:See also
553:,
541:,
533:) ≥ min(
529:,
513:,
509:,
458:) ≥ min(
390:) ≥ min(
291:) = max(
263:) = min(
162:interval
156:, where
143:function
914:Center
501:. Then
489:, i.e.
422:. Then
229:. The
160:is the
56:Please
857:
836:
599:T-norm
325:t-norm
936:2007.
474:)).
426:is a
412:group
279:)), (
242:union
141:is a
134:of a
855:ISBN
834:ISBN
557:)).
414:and
343:and
319:and
240:and
221:and
103:and
916:for
545:),
517:in
466:),
442:in
430:of
398:),
374:in
362:of
354:on
345:max
341:min
321:max
317:min
311:in
299:),
271:),
136:set
60:to
946::
521:,
446:,
438:,
378:,
370:,
337:ab
331:,
287:)(
259:)(
187:.
152:→
148::
127:A
115:.
555:z
551:y
549:(
547:R
543:y
539:x
537:(
535:R
531:z
527:x
525:(
523:R
519:X
515:z
511:y
507:x
503:R
499:X
495:X
491:R
487:X
483:R
472:y
470:(
468:A
464:x
462:(
460:A
456:y
454:*
452:x
450:(
448:A
444:G
440:y
436:x
432:G
424:A
420:G
416:A
408:G
404:y
402:(
400:A
396:x
394:(
392:A
388:y
386:*
384:x
382:(
380:A
376:X
372:y
368:x
364:X
360:A
356:X
333:b
329:a
313:X
309:x
305:x
303:(
301:B
297:x
295:(
293:A
289:x
285:B
281:A
277:x
275:(
273:B
269:x
267:(
265:A
261:x
257:B
253:A
249:B
245:A
238:B
234:A
227:X
223:B
219:A
209:L
185:A
181:L
174:L
158:L
154:L
150:X
146:A
139:X
132:A
83:)
77:(
72:)
68:(
54:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.