Fuzzy Entropy From Weak Fuzzy Subsethood Measures

Summary

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In this paper, we propose a new construction method for fuzzy and weak fuzzy subsethood measures based on the aggregation of implication operators. We study the desired properties of the implication operators in order to construct these measures. We also show the relationship between fuzzy entropy and weak fuzzy subsethood measures constructed by our method.

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Fuzzy Entropy From Weak Fuzzy Subsethood Measures

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1. Introduction

The fuzzy sets theory was introduced by Zadeh in 1965 (see [44]). Afterward he established the order relation between fuzzy sets in [45] as follows:

A ≤ B, if and only if µ^sub A^(x) ≤ µ^sub B^(x) for all x ∈ X,

where A and B are two fuzzy sets in the same universe X.

There have been many discussions about the non-fuzzy character of this order relation [1]. This fact led many authors (Bandler and Kohout [1], De Baets and Kerre [11], etc.) to propose different measures that provide an inclusion degree or subsethood measure of one fuzzy set in another. Mainly, three different axiomatizations of these fuzzy subsethood measures s have been given. The first one was given by Kitainik [25], the second one by Sinha and Dougherty [36], and the last one by Young [43].

These three axiomatizations have the first axiom (Axiom 1) in common:

(Axiom 1) σ (A, B) = 1 if and only if A ≤ B.

While Kitainik's axiomatization was the first one, it has not been widely studied until recently. On the other hand, Sinha and Dougherty (see [36]) proposed nine axioms any subsethood measure has to fulfill. They also introduced three axioms which depend on the application. In [12, 22] there is a study of the conditions in which Kitainik's axiomatization satisfies most of Sinha and Dougherty's axioms and vice versa.

The first axiom of Sinha and Dougherty is Axiom 1 and the second axiom is:

(Axiom 2) σ(A, B) = 0 if and only if ∃x ∈ X : µ^sub A^(x) - 1 and µB(?) = 0.

This axiom was criticized because one element in the referential determines the ...

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