Continuous fuzzy sets as probabilities of continuous fuzzy events

Abstract

In the first part of this study we explore continuous fuzzy numbers in the interval- and the α-cut forms to detect their similar nature. The conversion from one form to the other is a question of using the appropriate apparatus, which we also provide. Since the fuzzy numbers can reproduce fuzzy events we then will make a trial of extending the concept of fuzzy probability, defined by R. Yager [1] for discrete fuzzy events, on continuous fuzzy events. In order to fulfill the task we utilize conclusions made about fuzzy numbers to propose an initial conception of approximating the Gauss curve by a particularly designed function originated from the π-class functions. Due to the procedure of the approximation, characterized by an irrelevant cumulative error, we expand fuzzy probabilities of continuous fuzzy events in the form of continuous fuzzy sets. Furthermore, we assume that this sort of probability holds some conditions formulated for probabilities of discrete fuzzy events [2]-[7].