On Zadeh's Compositional Rule of Inference

This paper deals with Zadeh's compositional rule of inference [7] under triangular norms: IF X is P AND X and Y have relation R, THEN Y is Q, where P and Q are fuzzy sets of the real line IR, R is a fuzzy relation on IR and the conclusion Q is defined via sup-T composition of P and R: #Q (y) = sup x#IR T (# P (x), #R (x, y)), y # IR. It is shown that (i) when the triangular norm T and the membership function of the observation P are continuous, then the conclusion Q depends continuously on the observation; (ii) when T and the membership function of the relation R are continuous, then the observation Q has continuous membership function. Furthermore, we present a similar result for the discrete case 1 Preliminaries A fuzzy set A with membership function #A : IR # [0, 1] = I is called fuzzy number if A is normal, continuous,fuzzy convex and compactly supported. The fuzzy numbers will represent the continuous possibility distributions for fuzzy terms. Let A be a fuzzy number, th..