The Convexity of Fuzzy Sets and Augmented Extension Principle

Abstract

The classification of the fuzzy logic set is described as the fuzzy variables and fuzzy functions to be further interpreted in terms of the classical functions of modern analysis. Furthermore, the application of the extension principle is applied for the fuzzy functions here. There are theorems put in place to utilize the convexity principle to the fuzzy sets here. There are used definitions of the convexity and quasi-convexity of the fuzzy sets here. The decomposition theorem of the fuzzy set is stated in terms of the cuts of the fuzzy sets. Afterward, there was shown that the decomposition is applicable to the class of the fuzzy power functions. The extension principle is formulated to suggest that the fuzzy function and its inverse image of the fuzzy function can be fuzzified at the support plane of the fuzzy sets. The complexity of the problems exponentially emerges there, when the fuzzy sets are not convex. The theorems presented in this article are significant addendums to the extension principle to show that the extension principle may be drastically expanded in its application by utilization of the maximum or minimum support planes of the fuzzy set. The quasi-convexity feature of the fuzzy function is presented in the article here. This feature expands the extension principle to the level, when we can determine the arguments of the logic set to fuzzifying entire classical mathematical theories.