Characterizing rough fuzzy sets in constructive and axiomatic approaches

The theory of rough sets can be developed in at least two approaches: constructive and axiomatic approaches, which are complementary to each other. The constructive approach is more suitable for practical applications of rough sets, while the axiomatic approach is appropriate for studying the structures of rough set algebra. The constructive and axiomatic approaches are presented in the study of rough fuzzy set. In the constructive approach, one starts with a binary crisp relation and defines a pair of lower and upper rough fuzzy approximation operators. Different classes of rough fuzzy set algebra are obtained from different types of binary crisp relations. In the axiomatic approach, one defines a pair of dual rough fuzzy approximation operators and states that axioms must be satisfied by the operators. Various classes of rough fuzzy algebra are characterized by different sets of axioms. Axioms of rough fuzzy approximation operators guarantee the existence of certain types of binary crisp relations producing the same rough fuzzy operators.