(3.8)--3.10-AlgebraicstructureontheFuzz.pdf

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1、3.103.10 Algebraic Algebraic structure on structure on Determine whether Determine whether ,is is a)a)a a Boolean Boolean algebra.algebra.b)b)a a soft soft algebra.algebra.c)c)an an optimal soft algebra.optimal soft algebra.PosetPoset (,)Let Let,.,.Reflexivity Reflexivity ,.AntisymmetryAntisymmetry

2、,=,=.TransitivityTransitivity ,.Poset Poset(,)ReflexiveReflexive AntisymmetricAntisymmetric TransitiviteTransitivite Lattice Lattice(,)Suppose Suppose,=_,=_.Suppose Suppose ,is is anan upperupper boundbound ofof ,.,(),.,=.,=.,is a lattice.is a lattice.The The algebraic structure induced by the latti

3、ce algebraic structure induced by the lattice ,is is (,),wherewhere =,=,.=.=,=.=,=.()=(),()=().()=,()=.BoundednessBoundedness ,.=,=.=,=.CompletenessCompleteness Suppose Suppose is an identifier set,is an identifier set,.=,=._,_.,Distributive lawsDistributive laws ()=()(),()=()().Infinite Infinite di

4、stributive lawsdistributive laws ()=(),()=(),where where ,is an identifier set,is an identifier set,.Prove that Prove that ()=().Proof Proof ()=()=()=()=()=().ComplementationComplementation Suppose Suppose ，the complement of the complement of is is denoted by denoted by ,which membership function is

5、,which membership function is =,.Determine Determine whether the laws of complement whether the laws of complement hold in hold in ,.ExampleExample ，.Dual Dual lattice lattice Double Double negation law negation law ()=.De De Morgan lawsMorgan laws ()=,()=.Infinite Infinite De Morgan lawsDe Morgan l

6、aws ()=(),()=(),where where is an is an identifier identifier set,set,.Prove that Prove that ()=().()=()=()=()=()=()().DensityDensity is dense if for all is dense if for all,then,then there exists there exists ,such that,such that .Suppose Suppose,.,.Suppose Suppose,.DefineDefine =(+),.,.Determine whether Determine whether ,is is a)a)a a Boolean Boolean algebra.algebra.b)b)a a soft soft algebra.algebra.c)c)an an optimal soft algebraoptimal soft algebra.ConclusionConclusion ,is an optimal soft algebra.is an optimal soft algebra.,(,),(,)