模糊数学1 集合运算

1,BASICSLectureFuzzySet&Theory,OBJECTIVES1.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),2,OUTLINEII.BASICS,A.Definitionsandexamples1.Sets(集合的定义)2.Fuzzynumbers(模糊数的定义)B.Operationsonfuzzysetsunion(并),intersection(交),complement(补)C.Operationsonfuzzynumbersarithmetic,equations,functionsandtheextensionprinciple(扩展定理),3,DEFINITIONS,A.Definitions1.Setsa.Classicalsetseitheranelementbelongstothesetoritdoesnot.Forexample,forthesetofintegers,eitheranintegerisevenoritisnot(itisodd).However,eitheryouareintheUSAoryouarenot.WhataboutflyingintoUSA,whathappensasyouarecrossing?Anotherexampleisforblackandwhitephotographs,onecannotsayeitherapixeliswhiteoritisblack.However,whenyoudigitizeab/wfigure,youturnalltheb/wandgrayscalesinto256discretetones.,4,Classicalsets,Classicalsetsarealsocalledcrisp(sets).(紧集)Lists:A=apples,oranges,cherries,mangoesA=a1,a2,a3A=2,4,6,8,Formulas:A=x|xisaneven(偶的)naturalnumberA=x|x=2n,nisanaturalnumberMembershiporcharacteristicfunction(特征函数的隶属度),5,Definitionsfuzzysets,b.Fuzzysetsadmitsgradation(渐变)suchasalltones(色调)betweenblackandwhite(黑与白之间).Afuzzysethasagraphicaldescriptionthatexpresseshowthetransition(过渡)fromonetoanothertakesplace.Thisgraphicaldescriptioniscalledamembershipfunction(隶属函数).,6,Definitionsfuzzysets(figurefromKlir&Yuan),7,Definitions:FuzzySets(figurefromKlir&Yuan),8,Membershipfunctions(figurefromKlir&Yuan),9,Fuzzyset(figurefromEarlCox),10,FuzzySet(figurefromEarlCox),11,TheGeometryofFuzzySets(figurefromKlir&Yuan),12,Alphalevels,core,support,normal,13,Definitions:RoughSets,AroughsetisbasicallyanapproximationofacrispsetAintermsoftwosubsetsofacrisppartition,X/R,definedontheuniversalsetX.Definition:Aroughset,R(A),isagivenrepresentationofaclassical(crisp)setAbytwosubsetsofX/R,andthatapproachAascloselyaspossiblefromtheinsideandoutside(respectively)andwhereandarecalledthelowerandupperapproximationofA.,14,Definitions:Roughsets(figurefromKlir&Yuan),15,Definitions:IntervalFuzzySets(figurefromKlir&Yuan),16,Definitions:Type-2FuzzySets(figurefromKlir&Yuan),17,2.FuzzyNumber,AfuzzynumberAmustpossessthefollowingthreeproperties:1.Amustmustbeanormalfuzzyset,2.Thealphalevelsmustbeclosedforevery,3.ThesupportofA,mustbebounded.,18,1,Membershipfunction,isthesupport支ofz1isthemodalvalue重数,isana-levelof,a(0,1,a,FuzzyNumber(fromJorgedosSantos),a,19,1,Afuzzynumbercanbegivenbyasetofnestedintervals,thea-levels:,Fuzzynumbersdefinedbyitsa-levels(fromJorgedosSantos),.7,.5,.2,0,20,1,Triangularfuzzynumbers,21,FuzzyNumber(figurefromKlir&Yuan),22,B.OperationsonFuzzySets:UnionandIntersection(figurefromKlir&Yuan),23,OperationsonFuzzySets:Intersection(figurefromKlir&Yuan),24,OperationsonFuzzySets:UnionandComplement(figurefromKlir&Yuan),25,C.OperationsonFuzzyNumbers:AdditionandSubtraction(figurefromKlir&Yuan),26,OperationsonFuzzyNumbers:MultiplicationandDivision(figurefromKlir&Yuan),27,FuzzyEquations,28,ExampleofaFuzzyEquation(figurefromKlir&Yuan),29,TheExtensionPrincipleofZadeh,Givenaformulaf(x)andafuzzysetAdefinedby,howdowecomputethemembershipfunctionoff(A)?Howthisisdoneiswhatiscalledtheextensionprinciple(ofprofessorZadeh).Whattheextensionprinciplesaysisthatf(A)=f(A().Theformaldefinitionis:f(A)(y)=supx|y=f(x),30,ExtensionPrinciple-Example,Letf(x)=ax+b,31,再思想下列問題：,32,所謂模糊就是一種程度的問題(amatterofdegree),33,傳統集合論：設全集為,(iii),(iv)De-Morgan定律,(v)排中律：,34,模糊集合論：,35,（1）機率(probability)对未來的狀况无法完全知道；发生的随机性；因果律的残缺；发生上的不確定。,（2）模糊(fuz