Finite fuzzy sets

In this paper, we determine the number and nature of fuzzy subsets of a finite set of n elements taking membership values in the real unit interval. In order to do this we introduce an equivalence relation on the set of all fuzzy subsets. The important tools for studying this equivalence relation are that of a keychain, index of a keychain, and flags (maximal chains). These notions give rise to the idea of a pinned flag, their equivalences and an index of fuzzy subsets. Using these ideas we characterize the number of fuzzy subsets as the sum of a finite series of integral terms. These terms are enumerated by indices, each term representing the number of fuzzy subsets of a given index.