A Mathematical Theory of Fuzzy Numbers Granular Computing Approach

The essence of granular computing (GrC) is to replace the concept of points in classical mathematics by that of granules. Usual fuzzy number systems are obtained by using type I fuzzy sets as granules. These fuzzy number systems have a common weakness - lack of existence theorem. Let R be the real number system, the trapezoidal membership functions at r is an element of R is a base of fuzzified topological neighborhood system FNS(r). By taking FNS(r) as the granule, a new (but not type I) fuzzy number system F is formed. Surprisingly, we have found that such a new F is abstractly isomorphic to the classical real number system.