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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:

03E72 Theory of fuzzy sets, etc.
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