The generalized fuzzy derivative is interactive

Abstract

In this article, we prove that the generalized difference between A,B∈RFC, i.e., fuzzy numbers with continuous endpoints, is given by an interactive difference. To be more precise, we construct a certain joint possibility distribution I such that the generalized difference coincides with the sup-I extension of the subtraction. As an immediate consequence, we have that every notion of difference between A,B∈RFC, that has so far appeared in the literature, can be derived from a sup-J extension for some particular choice of J. Moreover, we show that both the generalized and the generalized Hukuhara derivative of a function f:R→RFC at x∈R can be expressed as the limit for h → 0 of a difference quotient, where the difference is an interactive difference for each h. For short, we say that the generalized (as well as the generalized Hukuhara) difference is interactive.