Fuzzy set

Fuzzy sets are an extension of classical set theory and are used in fuzzy logic. In classical set theory the membership of elements in relation to a set is assessed in binary terms according to a crisp condition — an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in relation to a set; this is described with the aid of a membership function. Specifically, a fuzzy set on a classical set mathbf{X} is defined as follows:

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ilde{mathit{A}}={(x,mu_{A}(x))mid xinmathbf{X}}

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The membership function mu_{A}(x) quantifies the grade of membership of the elements x to the fundamental set mathbf{X}. The value 0 means that the member is not included in the given set, 1 describes a fully included member (this behaviour corresponds to the indicator function of classical sets). Values strictly between 0 and 1 characterize the fuzzy members.

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Fuzzy set and crisp set

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The following holds for the functional values of the membership function mu_{A}(x)

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mu_{A}(x)ge0quadorallquad xinmathbf{X}

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sup_{xin X}=1

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This formulation of fuzzy set leads to the definition of fuzzy numbers and fuzzy intervals.

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A fuzzy number is a convex, normalized fuzzy set ilde{mathit{A}}subseteqmathbb{R}

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whose membership function is at least segmentally continuous and has the functional value mu_{A}(x)=1 at precisely one element.

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A fuzzy interval is a uncertain set ilde{mathit{A}}subseteqmathbb{R} with a mean interval whose elements possess the membership function value mu_{A}(x)=1. Likewise, the membership function must be convex

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and normalized and also at least segmentally continuous.

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The fuzzy set B, where B = {(3,0.3), (4,0.7), (5,1), (6,0.4)} would be enumerated as B = {0.3/3, 0.7/4, 1/5, 0.4/6} using standard fuzzy notation. Note that any value with a membership grade of zero does not appear in the expression of the set. The standard notation for finding the membership grade of the fuzzy set B at 6 is ?B(6) = 0.4.

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