University of Dortmund

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. 1 Some Fundamental Notions and Notations We fix a non-empty set U called universe. For arbitrary subsets X,Y j U we denote the union and the intersection of X andY by X ∪Y and X ∩Y , respectively. X is the complement of X with respect to U , i. e. the set U \ X where U \ X denotes the set-theoretical difference of U and X . The empty set is denoted by ∅. For crisp binary relations R, i. e. for R jU ×U , we use the terms “reflexive”, “symmetric”, “transitive”, “tolerance”, “semi partial ordering”, “equivalence” as usual. In doubt we refer to definition 2 where these notions are generalized to binary fuzzy relations. By U/R we denote the set of equivalence classes from U generated by the equivalence relation R on U . Assume R,R′ jU ×U . For definiteness we recall Definition 1 1. R ∪ R′ =def l[x,y] [x,y] ∈ R ∨ [x,y] ∈ R′q 2. R ° R′ =def l[x,y] ∃z([x,z] ∈ R ∧ [z,y] ∈ R′)q 3. R0 =def k[x,x] x ∈ Up 4. Rn+1 =def R ° Rn where n is an integer with n ≧ 0 ∗Revised version of a paper originally published in 5th European Congress on Intelligent Techniques & Soft Computing (EUFIT ’97), Aachen, Germany, September 8–11, 1997, vol. 1, pages 159–167