Representation of fuzzy operators using ordinary sets

A novel interpretation of union, intersection, and inclusion in fuzzy sets in the light of measure theory is given. The existing definitions of these operators are based only on the value of membership functions characterizing fuzzy sets. The proposed definitions take into account the nature (behavior) of the membership functions together with their values. The uniqueness of the proposed definition is established. These definitions are generalized for any arbitrary continuous function defined on any bounded closed interval. The existing definitions can also be derived from the framework proposed.