Initial Comparison of Formal Approaches to Fuzzy and Rough Sets

Fuzzy sets and rough sets are well-known approaches to incomplete or imprecise data. In the paper we compare two formalizations of these sets within one of the largest repositories of computer-checked mathematical knowledge – the Mizar Mathematical Library. Although the motivation was quite similar in both developments, these approaches – proposed by us – vary significantly. Paradoxically, it appeared that fuzzy sets are much closer to the set theory implemented within the Mizar library, while in order to make more feasible view for rough sets we had to choose relational structures as a basic framework. The formal development, although counting approximately 15 thousand lines of source code, is by no means closed – it allows both for further generalizations, building on top of the existing knowledge, and even merging of these approaches. The paper is illustrated with selected examples of definitions, theorems, and proofs taken from rough and fuzzy set theory formulated in the Mizar language.