Abstract
In the area of fuzzy set theory, one of the important themes is how to extend Dempster-Shafer (D-S) theory to include the process of fuzzy events. In this paper, instead of extending D-S rules, we propose a direction of arguing that elements in a fuzzy domain are all singletons based on the statistical viewpoint and are independent of one another. Therefore, the D-S combination rule can consequently be applied as regularly as in non-fuzzy cases. We first examine the characteristics of how membership functions are defined, which leads to the conclusion that every element in a fuzzy domain is independent, mutually exclusive of each other and, consequently, a singleton. We then discuss the relationships existing between the grades of a membership function and its corresponding statistical probability, and between the probability of a fuzzy event and the probability of every element in its associated data domain. The equations are then derived to calculate the probability of every element in a fuzzy event data domain when both the probability of the event and its membership functions are known. An example is finally given to illustrate discussions.
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