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1、BASICS–LectureFuzzySet&TheoryOBJECTIVES1.Todefinethebasicideas(概念)andentities(本质)infuzzysettheory(模糊集合理论)2.Tointroducetheoperationsandrelationsonfuzzysets(模糊集合)3.Tolearnhowtocomputewithfuzzysetsandnumbers(模糊集合与模糊数)-arithmetic(计算),unions(并),intersections(交),complements(补)1OUTLINE II
2、.BASICSA.Definitionsandexamples1.Sets(集合的定义)2.Fuzzynumbers(模糊数的定义)B.Operationsonfuzzysets–union(并),intersection(交),complement(补)C.Operationsonfuzzynumbers–arithmetic,equations,functionsandtheextensionprinciple(扩展定理)2DEFINITIONSA.Definitions1.Setsa.Classicalsets–eitheranelementbelon
3、gstothesetoritdoesnot.Forexample,forthesetofintegers,eitheranintegerisevenoritisnot(itisodd).However,eitheryouareintheUSAoryouarenot.WhataboutflyingintoUSA,whathappensasyouarecrossing?Anotherexampleisforblackandwhitephotographs,onecannotsayeitherapixeliswhiteoritisblack.However,whe
4、nyoudigitizeab/wfigure,youturnalltheb/wandgrayscalesinto256discretetones.3ClassicalsetsClassicalsetsarealsocalledcrisp(sets).(紧集)Lists:A={apples,oranges,cherries,mangoes}A={a1,a2,a3}A={2,4,6,8,…}Formulas:A={x
5、xisaneven(偶的)naturalnumber}A={x
6、x=2n,nisanaturalnumber}Membershiporcharac
7、teristicfunction(特征函数的隶属度)4Definitions–fuzzysetsb.Fuzzysets–admitsgradation(渐变)suchasalltones(色调)betweenblackandwhite(黑与白之间).Afuzzysethasagraphicaldescriptionthatexpresseshowthetransition(过渡)fromonetoanothertakesplace.Thisgraphicaldescriptioniscalledamembershipfunction(隶属函数).5Defin
8、itions–fuzzysets(figurefromKlir&Yuan)6Definitions:FuzzySets(figurefromKlir&Yuan)7Membershipfunctions(figurefromKlir&Yuan)8Fuzzyset(figurefromEarlCox)9FuzzySet(figurefromEarlCox)10TheGeometryofFuzzySets(figurefromKlir&Yuan)11Alphalevels,core,support,normal12Definitions:RoughSetsArou
9、ghsetisbasicallyanapproximationofacrispsetAintermsoftwosubsetsofacrisppartition,X/R,definedontheuniversalsetX.Definition:Aroughset,R(A),isagivenrepresentationofaclassical(crisp)setAbytwosubsetsofX/R,andthatapproachAascloselyaspossiblefromtheinsideandoutside(respectively)andwhereand
10、arecalledtheloweranduppera
