Aggregation of the fuzzy logic sets in terms of the functions of the triangular norm and triangular co-norm

Abstract

The purpose of this article is to explore the group of the operators, which can be used for aggregation of the fuzzy sets. There were scrutinized operators to be used for the intersection and union such as the triangular norm and triangular co-norm in the article. All of those operators are defined as the binary operations, where there was indicated that the norms and co-norms are two-valued functions of [0, 1]. Furthermore, there were discussed properties of norm and co-norm functions such as symmetry, associativity, neutral entry and null entry properties, monotonicity properties. Moreover, the consideration of the properties of norm and co-norm functions have routed investigation of the norm and co-norm operators to go further to prove theorem, which converts the union into intersection and vice versa in terms to all possible norm and co-norm functions based on the fuzzy logic sets.

The advanced defuzzification methods of the convex α – cut fuzzy sets

“…Definition. The elements of the fuzzy set where the membership function approaches the value to the certain degree of 𝛼 is called the 𝛼-cut of the fuzzy set [13,14]:…”

“…Then, according to the definition of the 𝛼-cuts we can represent the fuzzy sets as stated above: The fuzzy set is convex if all its cuts are convex in terms of the convexity. We can re-define the cuts of the fuzzy sets by the pair of the fuzzy set with reference to their lowest upper bound and greatest lower bound defined as it is [13,14,15,16,17,18,19].…”

“…One of the methods of the defuzzification is the 𝛂-cut method. Here for the given fuzzy set 𝐹(𝑥) we are going to define the crisp set 𝐹 𝛼 (𝑥) = {𝑥:𝜙 𝐹 𝛼 (𝑥) ≥ 𝛼, 0 < 𝛼 < 1 [13,14,15].…”

“…The membership function of the fuzzy set 𝐹 ⊆ 𝑋 is denoted as 𝜙 𝐹 (𝑥) and defined as it is in [13,14,15] which can be interpreted by the following: nearer the value of the grade of membership to 1 then 𝑥 is more affiliated to 𝐹 :…”

“…There is the following theorem which states that for the convex fuzzy sets there is existing the defuzzied value of the argument of the aggregated crisp function [13,14].…”