Semantics of fuzzy sets in rough set theory

The objective of this paper is to provide a semantic framework for fuzzy sets in the theory of rough sets. Rough membership functions axe viewed as a special type of fuzzy membership functions interpretable using conditional probabilities. The relationships between fuzzy membership functions and rough membership functions, between core and support of fuzzy set theory and lower and upper approximation of rough set theory, are investigated. It is demonstrated that both theories share the same qualitative properties. Interpretations of fuzzy sets in rough set theory lead to constraints on membership values. Two types of constraints on membership values are studies, namely, constraints on membership values of related elements and constraints on membership values of related sets. The classical rough set model and generalized rough set models are discussed.