Fuzzy sets.(English)Zbl 0139.24606

A fuzzy set is a “set” of elements with a continuum of “grades of membership”. The rigorous definition is: let $$X$$ be a set of objects (elements); a fuzzy set $$A$$ in $$X$$ is defined by a “membership (characteristic) function” $$f_A$$, which associates with each element $$x\in X$$ a real number $$f_A(x)\in [0,1]$$. The value $$f_A(x)$$ of $$f_A$$ at $$x$$ represents the grade of membership of $$x$$ in $$A$$. If $$A$$ is an “ordinary” set, its membership function $$f_A$$ can take on only the values 0 and 1: $$x\in A\Leftrightarrow f_A(x) = 1$$ and $$x\neq A \Leftrightarrow f_A(x)=0$$. Various usual notions are extended to such sets (for instance, the notions of union, intersection and convexity).

MSC:

 3e+72 Theory of fuzzy sets, etc.
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References:

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