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A new method for translating a fuzzy rough set to a fuzzy set is introduced and the fuzzy approximation of a fuzzy rough set is given. The properties of the fuzzy approximation of a fuzzy rough set are studied and a fuzzy entropy measure for fuzzy rough sets is proposed. This measure is consistent with similar considerations for ordinary fuzzy sets and is the result of the fuzzy approximation of fuzzy rough sets.

This paper presents the concepts of isomorphism and homomorphism of fuzzy sets (fuzzy relation), gives sufficient and necessary conditions of isomorphism and homomorphism of fuzzy sets (fuzzy relation). Fortunately, we point that the essence of fuzzy sets (fuzzy relations) in X ,which give the order relation of elements in X(XtimesX).As long as the corresponding order relations are consistent, conclusions from any fuzzy membership function(fuzzy relation) defined by the order relation are certainly same. We propose the concept of classified isomorphism of the fuzzy relation and prove that if two fuzzy similarity relation are isomorphic, then they must be isomorphic classified by the concept of isomorphic classification of fuzzy relation.

This paper discussed the relation between fuzzy sets and rough sets by theory of the isomorphism and homomorphism of fuzzy sets, and two main conclusions are reached. Firstly, for any group of fuzzy sets which are isomorphic for each other in X, an approximation space could be defined uniquely. Secondly, for any group of fuzzy equivalence relations which are isomorphic, similarly a group of fuzzy approximation spaces which are isomorphic could be defined. Moreover, the construction of fuzzy approximation space is given. Finally, we illustrate an concrete example to show the construction of the fuzzy approximation space, and discussed their related properties.