Theory Based on Interval-Valued Level Cut Sets of Zadeh Fuzzy Sets

Abstract

In this paper, the concepts of interval-valued level cut sets on Zadeh fuzzy sets are presented and new decomposition theorems of Zadeh fuzzy sets based on new cut sets are established. Firstly, four interval-valued level cut sets on Zadeh fuzzy sets are introduced, which are generalizations of the normal cut sets on Zadeh fuzzy sets and have the same properties as that of the normal cut sets on Zadeh fuzzy sets. Secondly, based on these new cut sets, the new decomposition theorems of Zadeh fuzzy sets are established. It is pointed that each kind of interval-valued level cut sets corresponds to two decomposition theorems. Thus eight decomposition theorems are obtained. Finally, the definitions of ¯ L-inverse order nested sets and ¯ L-order nested sets are introduced and we established eight new representation theorems on Zadeh fuzzy sets by using the concept of new nested sets.