Operations on Fuzzy Sets

Definitions of set-theoretic operations such as the complement, union and intersection can be extended from ordinary set theory to fuzzy sets. As membership degrees are no longer restricted to 69#69 but can have any value in the interval 70#70, these operations cannot be uniquely defined. It is clear, however, that the operations for fuzzy sets must give correct results when applied to ordinary sets (an ordinary set can be seen as a special case of a fuzzy set). This section presents the basic definitions of fuzzy intersection, union and complement, as introduced by Zadeh. General intersection and union operators, called triangular norms (71#71-norms) and triangular conorms (71#71-conorms), respectively, are given as well. In addition, operations of projection and cylindrical extension, related to multi-dimensional fuzzy sets, are given.