模糊数学Ch1_Sec3-4-5

1.3t-normandt-conorm(p.10),Problemandoffuzzysetsaredefinedbyoperatorsand,But0.8=0.80,0.8Informationislost.,TriangleNorm(Menger,1942),Definition1.3.1/1.3.2Amapping:0,10,10,1issaidatrianglenorm,ifitsatisfiesthefollowingconditionsCommutativity(a,b)=(b,a);Associativity(a,b),c)=(a,(b,c);Monotonyac,bd(a,b)(c,d)Tist-norm=trianglenorm+T(1,a)=aSist-conorm=trianglenorm+S(a,0)=aT(a,b)andS(a,b)arewrittenasaTbandaSbrespectively.,Commonoperator,Namet-conorm(S)t-norm(T)ZadehDrasticAlgebraBoundedEinsteinHamacherseebookYager,Theorem1.3.3ForallTandSTSTheorem1.3.8Leta,b0,1,then(1)(2)(3)(4)Proof(1),1.4MeasurementofFuzziness(p.29),1.4.1Cardinalityoffuzzysetse.g.A=1,2,3,4HowmanyelementsarethereincrispAAnswer:4CardinalityofAis4.,基数或势,Definition1.4.1ForfiniteuniverseXCardinalityoffuzzysetARelativecardinalityoffuzzysetA,基数或势,1.4.2Measureoffuzzinessonfuzzysets(p.29),Whatdegreeoffuzzinessisforafuzzyset(LucaA(x)1/2d(A)=1;(xX)B(x)A(x)1/2d(B)d(A);AF(X),d(A)=d(Ac).ThendissaidmeasureoffuzzinessinF(X).,模糊度,经典集不模糊,这样的集最模糊,越靠近1/2越模糊,对称性,Examples(p.32),HammingEuclidWhereShannonWhere,0,1,1/2,ln2,e.g.1.4.3LetX=a,b,c,dandThen,.,1.4.3Crispdomainoffuzzyoperator(p.33),A(x)=0orA(x)=1xisacrisppoint.A(x)(0,1)xisafuzzypoint.Definition1.4.3Let*beafuzzyoperatoron0,1andThendomain(*)issaidthecrispdomainoffuzzyoperator*.,清晰域,清晰点,e.g.()=(x,y)|xy=0or1=(x,y)|x=0ory=0(x,y)|x=y=1()=(x,y)|max(0,x+y-1)=0or1=(x,y)|x+y11,1,1.4.5Degreeofsimilarityoffuzzysets(p.36),Degreeofsimilarityisasortofmeasurefortwosetsontheirdegreeofcloseness.Definition1.4.5LetA,B,CF(X)andamappingN:F(X)F(X)0,1satisfythefollowingconditions:N(A,B)=N(B,A);N(A,A)=1,N(X,)=0;ABCN(A,C)N(A,B)N(B,C).ThenNissaidmeasurefunctionofsimilarity.,贴近度,对称性,最贴近与最不贴近的情形,离的越远贴近度越小,1HammingNH2EuclidNE,3Max-minNM,4Min-averageNA,e.g.1.4.6Let,Then,and,1.4.6Latticemeasureofsimilarityoffuzzysets(p.39),Latticedegreeofsimilarityisanothersortofmeasurefortwosetsontheirdegreeofcloseness.Definition1.4.5LetA,BF(X)and,ThenandarerespectivelysaidinnerproductandouterproductoffuzzyAandB.,Definition1.4.8LetA,BF(X),thenissaidalatticemeasureofsimilarityofAandBonF(X),NL(A,B)iscalledalatticemeasurefunctionofsimilarityofAandBonF(X).,格贴近度函数,格贴近度,e.g.InExample1.4.6,X=x1,x2,x3,x4,x5,x6andAndthatAB=0.9,NL(A,B)=0.9(1-0.5)=0.5,1.5ExtensionofFuzzysets(p.42),1.5.1type2fuzzysetsx0.8xolderDefinition1.5.1(Mizumoto&Tanaka,1976)AmappingA:XF(0,1)issaidatype2fuzzysetA.e.g.1.5.1LetA=“oldness”Type2fuzzysetA=,A:XF(0,1),1.5.2Interval-valuedfuzzysets(p.44)x0.8x0.7,0.9Definition1.5.3(Sambuc,1975)AmappingA:XI0,1issaidaninterval-valuedfuzzyset.WhereI0,1isallintervalson0,1.e.g.Ine.g.1.5.1LetA=“oldness”,A:XI0,1,1.5.4Intuitionisticfuzzysets(p.45)Definition1.5.5(Atanassov,1983)Theobjectissaidanintuitionisticfuzzyset.WhereismembershipdegreethatxbelongstoA.ismembershipdegreethatxdonotbelongtoA.,直觉模糊集,e.g.Ine.g.1.5.1，letX=x1,x2,x3,x4,x5thenA=“oldness”isanintuitionisticfuzzyset.,upperapproximation,lowerapproximation,subsetX,1.5.5Roughsets(p.47),粗糙集,Definition1.5.9(Pawlak,1982)LowerapproximationUpperapproximation,Example1.5.5GiventoybrickssetU=x1,x2,x8withdifferentcolor(red,yellowandblue),shape(square,roundandtriangle)andbulk(littleandgreat)Thereforethesetoybricksaredescribedbycolor,shapeandbulk.Forexample,atoybrickmaybered,littleandround,oryellow,greatandsquare.Ifwedescribethesetoybricksbyanattribute,thentheymaybeclassifiedbycolor,shapeandbulk.,Classificationbycolor:x1,x3,x7red；x2,x4blue；x5,x6,x8yellowClassificationbyshape:x1,x5round；x2,x6square；x3,x4,x7,x8triangleClassificationbybulk：x2,x7,x8great；x1,x3,x4,x5,x6little,Inotherwords,wedefinethreeequivalentrelations(i.e.attributes):color(R1)，shape(R2)andbulk(R3).Bytheseequivalencerelationswehavethreeequivalentclassesasfollows:,Classificationbycolor:x1,x3,x7red；x2,x4blue；x5,x6,x8yellowx1,x3,x7x