Fuzzy Rough Sets versus Rough Fuzzy Sets

Abstract

The starting point of the paper is the (well-known) observation that the classical Rough Set Theory as introduced by PAWLAK is equivalent to the S5 Propositional Modal Logic where the reachability relation is an equivalence relation. By replacing this equivalence relation by an arbitrary binary relation (satisfying certain properties, for instance, reflexivity and transitivity) we shall obtain generalized (crisp!!) rough set theories. Our ideas in the paper are: 1. We replace the crisp reachability relation by a binary fuzzy relation whereas the set to be approximated remains crisp. It is very important that the reachability relation is used as a fuzzy relation, i. e. without introducing and using a cut point. Hence, these lower and upper fuzzy approximations of the given crisp set are fuzzy sets, in general. 2. Vice versa, the given set to be approximated is a fuzzy set, but the reachability relation is crisp. Also in this case the lower and the upper crisp approximations of the given fuzzy set are again fuzzy sets, in general. 3. Finally, we define a lower and an upper approximation of a fuzzy set using a binary fuzzy relation. It is interesting that this approach coincides with a concept which we have developed for interpreting the modal operators Box and Diamond in the framework of Fuzzy Logic.