On the Analytic Formalism of the Theory of Fuzzy Sets

The extension of standard concepts of set theory (like union, intersection, etc.) to the theory of fuzzy sets is not obvious. Several policies for answering questions of this type have been discussed, but it appears that the operations of max and min play a central role in the arithmetic of fuzzy sets. Letting Pi denote our willingness to accept x as a member in the fuzzy set A, Zadeh defines intersection A fl B and union A U B of the two fuzzy sets A and B by A n B = 1(x; min M4, p&)J)l and A U f? = {(x; max &&Y), &W. The object of this paper is to show that these definitions are not only natural, but under quite reasonable assumptions the only ones possible.