Compatibility between fuzzy set operations and level set operations: Applications to fuzzy difference

Three types of binary operation between any two fuzzy sets are proposed in this paper. The concept of 0-level set of fuzzy set plays an important role for defining the binary operations between two fuzzy sets. According to the decomposition theorem of fuzzy sets, the consistency between the membership functions and α-level sets should be rigorously verified when the binary operation between two fuzzy sets is defined by using their α-level sets. The consistency is investigated thoroughly in this paper for the purpose of guaranteeing the existence of such binary operations. Moreover, we also propose the so-called Hausdorff difference between any two fuzzy sets to define the difference of any two fuzzy numbers.