Duality in Fuzzy Sets and Dual Arithmetics of Fuzzy Sets

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.

Keywords

• dual decomposition theorem
• dual fuzzy sets
• dual arithmetics
• lower α-level sets
• upper α-level sets