模糊数学Ch2_Sec1-2-3-4

CH2DecompositionTheorem,RepresentationTheoryandExtensionPrinciple,2.1Cutsetsoffuzzysetse.g.2.1.1Inanexamabout“winner”,marksoftenexamineesareasfollows.Examineex1x2x3x4x5x6x7x8x9x10Grade(mark)100943261855525728640Nowwepickoutwintersaccordingtoprinciple“enrollingonlythosewhoareoutstanding”.,PickoutelementswhichmembershipdegreesatisfiesFuzzysetCrispset,Definition2.1.1LetAF(X)and0,1.-cutsetoffuzzysetAor-levelsetofA;strong-cutsetorstrong-levelset；thresholdvalueorbelieflevel.,截集,阈值,置信水平,e.g.2.1.2Letusconsiderexample1.2.3again:A=“round”=,e.g.2.1.3LetAF()and,Theorem2.1.1LetA,BF(X)and,0,1,then(1)(2)e.g.,1,Afuzzysetcanbegivenbyasetofnestedintervals,thea-levels:,Fuzzysetsdefinedbyitsa-levels,.7,.5,.2,0,Theorem2.1.1LetA,BF(X)and,0,1,then(1)(2)ProofOnlyprove(2)“”IfAB,thenforall0,1,Namely,(2),ForwehavebutThisiscontradictionwithhypothesis.,“”Letforall0,1.IfABisnotrue,thenthereexistx0Xsuchthat,Theorem2.1.4LetA,BF(X),then(1)(2)ProofTheotheridentitiescanbeprovedbysimilarmethod.,，,.,Theorem2.1.6LetandAF(X),then(2)Proof(2)Thesecondidentitycanbeprovedbysimilarmethod.,.,；,.,GenerallyExample2.1.5LetThenAndTherefore,；,，,2.2DecompositionTheorem(P56),e.g.2.1.110kerAsuppA,Definition2.2.1LetAF(X)and0,1.WedefineafuzzysetAwithmembershipasAissaidscalarproduct(数积)andA.IfAisacrispset,then,Theorem2.2.2LetAF(X),then,Theorem2.2.2LetAF(X),thenProof,Corollary2.2.1LetAF(X),thenforallxXProofx的隶属度=包含x的所有截集中的最大值。,分解定理,e.g.2.2.1Letandforall0,1?,2.3RepresentationTheory(P59),2.4ExtensionPrinciple(P66),XYF(X)F(Y)2.4.1UnaryExtensionPrinciple(一元扩张原理)LetmappingThenitisinducedanewmapping,stillwrittenasf,f:P(X)P(Y)Where,yf(x)f(A)0Ax,Theorem2.4.1Letf:XYandAP(X),thenforallyY,Definition2.4.1(ExtensionPrincipleI)Letmappingf:XY,Thenitisinducedthefollowingmapping,writtenasff:F(X)F(Y),Af(A)Where,Example2.4.1Letand,，,x1x2x3x4x5x6,Example2.4.1x1x2x3x4x5x6,，,2.4.2MultivariateExtensionPrinciple(多元扩张原理),Theorem2.4.11Let,andF(Xi)then,Example2.4.4LetandThen,，,Example2.4.4Let,，,3=1+2=2+1=3+0=0+3=,ItisremarkablethatoperationsinExample2.4.4canberegardedasgenerali