Interpreting Fuzzy Membership Functions in the Theory of Rough Sets

Abstract

A fundamental difficulty with fuzzy set theory is the semantical interpretations of membership functions. We address this issue in the theory of rough sets. Rough membership functions are viewed as a special type of fuzzy membership functions interpretable using conditional probabilities. Rough set approximations are related to the core and support of a fuzzy set. A salient feature of the interpretation is that membership values of equivalent or similar elements are related to each other. Two types of similarities are considered, one is defined by a partition of the universe, and the other is defined by a covering.