Abstract
The conventional concept of α-level sets of fuzzy sets will be treated as the upper α-level sets. In this paper, the concept of lower α-level sets of fuzzy sets will be introduced, which can also be regarded as a dual concept of upper α-level sets of fuzzy sets. We shall also introduce the concept of dual fuzzy sets. Under these settings, we can establish the so-called dual decomposition theorem. We shall also study the dual arithmetics of fuzzy sets in R and establish some interesting results based on the upper and lower α-level sets.
Highlights
The properties of fuzzy numbers and the arithmetic of fuzzy quantities have been studied for a long time
We are going to establish the so-called dual decomposition theorem by showing that the membership function can be expressed in terms of lower α-level sets
Let à be a fuzzy subset of a universal set U with membership function denoted by ξ Ã
Summary
The properties of fuzzy numbers and the arithmetic of fuzzy quantities (or fuzzy numbers) have been studied for a long time. We shall study the dual arithmetic of fuzzy sets by considering the dual membership function. We are going to establish the so-called dual decomposition theorem by showing that the membership function can be expressed in terms of lower α-level sets. The concept of dual fuzzy set will be proposed by considering theso-called dual membership function. Based on the dual membership functions, we shall study the so-called dual arithmetic of fuzzy sets in R. Inspired by its form, we shall define the so-called dual arithmetic operations based on the infimum and maximum of dual membership functions. R and establish some interesting results based on the upper and lower α-level sets
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