Fuzzy sets and membership functions based on probabilities

Since the introduction of fuzzy set theory in 1965, several attempts to establish the relationship between the grades of membership and the classical probability measures have been made. It turns out that there are different sources of fuzziness that must be dealt with differently. In the present paper we examine in detail two types of fuzziness, namely, the fuzziness due to classification in an under-or overdimensioned universe and the fuzziness due to the intersubject differences in opinion. For the former case the membership function is defined to be equal to the normalized distance from the point to the boundary of the set in a specific metric. It is shown that this definition of the membership function is fully consistent with the max-min operations for the union/intersection; however, the membership function of the complement is defined differently from the usual "one minus" rule. The fuzziness due to the intersubject differences turns out to be a simple averaging process, and the explicit formulas for this case were derived. Several examples that illustrate the notions of fuzzy intervals and fuzzy numbers are given and the interpretations of the derived membership curves are presented. The formulas for calculation of the membership function of a sum of fuzzy numbers and of a product of fuzzy numbers times a constant are derived. An extended definition of the measure of fuzziness is presented and applied to the defined membership functions.