The Study Of Some Topics On Vague Sets

To compensate for the limitations of the fuzzy set theory in dealing with uncertain information, Gau and Buehrer proposed Vague set in1993, which is a new tool for dealing with fuzzy information. Vague set further develop the theory of fuzzy set, and it can accurately describe and depict the essence of fuzziness of the objective world. In this paper, based on decomposition theorem, representation theorem, extension principle of fuzzy sets, we consider the true and false memberships of fuzzy sets as a number pair μ=(μt,μf)by introducing the concept of binary cut sets of Vague sets. Through this concepts, decomposition theorem, representation theorem and extension principle of Vague sets are described and the multiple extension principle of Vague sets are obtained. The concept of direct product of vague set is given by using representation theorem of vague set and multi-extension principle of vague sets is obtained. Finally, Based on the existent conclusions, we give a new definition of the product μ nd A, we define:tμA(u)=μtvtA(u),1-fμA(u)=μfv(1-fA (u)), for any u∈U, denoted by μ°A. Based on μ°A, a new decomposition theorem, representation theorem, extension principle and multi-expansion theory of Vague sets are obtained. These new results clearly express the relationships between Vague sets and classical sets and develop theories of Vague set.