Generalized fuzzy sets and fuzzy relations

Abstract

In classical fuzzy set theory, a fuzzy set is a membership function which associates with each element a real number in [0, 1], a fuzzy relation is a function which associates with each pair of elements a real number in [0, 1]. In the present paper, we generalize the above two concepts by associating with each set a real number in [0, 1], and associating with each pair of sets a real number in [0, 1], respectively. We then give a series of properties for these two types of generalized concepts. We also show that a generalized fuzzy relation can be induced by a classical fuzzy relation, which shows the communication of our generalized fuzzy concepts with the classical fuzzy theory.