an:03226241
Zbl 0139.24606
Zadeh, L. A.
Fuzzy sets
EN
Inf. Control 8, 338-353 (1965).
00134593
1965
j
03E72
fuzzy sets; continuum of grades of membership; membership characteristic function; separation theorem for convex fuzzy sets
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).
N. C. A. da Costa