The Fuzzy Description of the Boundary Region in Rough Sets and its Applications

In this paper, the boundary region of a rough set in the universe U is divided into lower and upper boundaries, and a fuzzy set is defined in the boundary region of the rough set based on the theory of inclusion degree. It can descript the element in boundary region of any rough set more precision; therefore, it can characterize the process which the rough set from vagueness to crisp exactly, and it also shows the nature of the rough set. Based on this definition, we redefine the rough membership function by the fuzzy set on the boundary region, and the operation of the intersection and union of any two fuzzy set will be also hold under the new rough membership function. Hence, the deficiency of the works by Pawlak (1985) and Sun Bing zhen and Gong Zeng tai (2006) is overcome. Finally, the measure of fuzziness of any rough set is discussed by the theory of fuzzy entropy.