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1、4.24.2 Properties of Fuzzy OperatorsProperties of Fuzzy Operators :,BoundaryBoundary conditionsconditions fuzzy operator =,=t-norm ,=s-norm ,=s s-normnorm t t-normnorm Fuzzy OperatorFuzzy Operator generalized maximumgeneralized maximum generalized generalized minimumminimum Example 1Example 1 Determ
2、ine whether Determine whether is a tis a t-norm.norm.SolutionSolution a)a)=.b)b)=.c)c)If If ,then,then .d)d)=.Example 2Example 2 Determine whether Determine whether is is an san s-normnorm.SolutionSolution a)a)=.b)b)=.c)c)If If ,then then .d)d)=.Example 3Example 3 Determine whether a Determine wheth
3、er a t t-normnorm is a fuzzy operator.is a fuzzy operator.AnalysisAnalysis ,=,=.,=,=.Example Example 4 4 Determine whether an Determine whether an s s-normnorm is a fuzzy operatoris a fuzzy operator.AnalysisAnalysis ,=,=.,=,=.AnalysisAnalysis Suppose Suppose is a is a t t-normnorm,.,=,=,=,=.,=.,=.So
4、 So is a fuzzy operatoris a fuzzy operator.ConclusionConclusion Let Let be be a ta t-normnorm,be be an san s-normnorm.Then Then both both and and are are fuzzy fuzzy operatorsoperators,and,and ,=,=.Weaker Weaker thanthan Let Let,be fuzzy operators.If be fuzzy operators.If,then then is is weakerweaker thanthan,denoted by,denoted by .Theorem Theorem 1 1 Let Let be a tbe a t-norm,norm,be be an an s s-norm.Then norm.Then .Proof Proof Suppose Suppose,.ProveProve that that .,=,=,.ProveProve thatthat .,=,=,.ProveProve thatthat .,.:,BoundaryBoundary conditionsconditions =,=.=,=.,=,=.,=,=.