The Convex Fuzzy Sets and Their Properties with Application to the Modeling with Fuzzy Convex Membership Functions

Abstract

The convex fuzzy sets are discussed in terms of the extension principle of Fuzzy logic sets in cases, when characteristic function is not one-to one. The new fuzzifying extension theorem is introduced to apply the concepts of the fuzzy sets to the non-convex fuzzy sets. This theorem enables to find the crossover points of the fuzzy sets, which gives new outlook to the modeling of the non-convex fuzzy sets in terms of the convexity of the fuzzy sets. The power fuzzy convex sets and their properties are established in the article here to show the convexity of all level cuts of the sub-sets of the fuzzy set. There were presented the properties of the quasi-convex fuzzy sets, which indicates the support plane of the fuzzy sets, where the characteristic function is quasi-convex at the support point. Furthermore, there were introduced the properties of the quasi-convex fuzzy sets, where the convexity of the fuzzy sets can be achieved at the mid-point of the suggested interval of uncertainty, where the support point is located. There were presented the properties of the quasi-convex fuzzy sets, where the extension principle to apply the fuzzy categories can be utilized. There were presented the various types of the constructed convex membership functions based on the new convexity properties established in the article here.