The new approach to the construction of parametric membership functions for fuzzy sets with unequal supports

摘要

The current research is devoted to developing methods of a novel mathematical interpretation of term-sets of linguistic variables. To the term-sets of the linguistic variables fuzzy sets are assigned. We intend to adopt the π -functions and the π -functions to derive formulas of membership functions of these sets. The fuzzy sets are divided in three families in the case of an odd number of the term-sets. To each family, we assign only one parametric formula, which depends on two parameters: the width of a non-fuzzy set, which contains all supports of the fuzzy sets being representatives of the term-sets, and a number of the term-sets. Provided that the supports of fuzzy sets will be unequal, the membership function of the set, belonging to one of the families, is computed by means of a functional modifier, inserted in the common equation typical of this family. Medical examples explain how to use cumulated membership functions practically. The procedure can be easily computerized.