Complex fuzzy sets: (r, θ)-cut sets, decomposition theorems, extension principles and their applications1

Abstract

Complex fuzzy set, as an extension of classical fuzzy sets, could describe the fuzzy characters of things more detail and comprehensively and is very useful in dealing with vagueness and uncertainty of problems that include the periodic or recurring phenomena. Note that a complex fuzzy set is different from the fuzzy complex set introduced and discussed by many scholars, since the membership degree of a complex fuzzy set is a complex number with length less than or equal to 1 while a fuzzy complex set is a real number with membership degree less than or equal to 1, and the universe is the complex plane. As the mathematical theoretical basis of fuzzy mathematics, fuzzy set and its mapping, corresponding fuzzy complex set and its mapping have been investigated in depth because they integrate and cross the methods and results of classical real analysis and complex analysis. However, there is no comprehensive investigation on complex fuzzy set and its corresponding mathematical theory, even include decomposition theorems, extension principles and the basic operations of the complex fuzzy set. As is well known, the cut set of fuzzy sets is the bridge between fuzzy sets and classical sets, which plays a significant role in fuzzy sets and fuzzy systems. In this paper, the concept of (r, θ)-cut sets of complex fuzzy sets is proposed and their properties are discussed. Meanwhile, the decomposition theorems and the extension principles of complex fuzzy set based on (r, θ)-cut sets are deduced and corresponding properties are investigated. All these conclusions not only deeply enrich the fundamental theory of complex fuzzy set, but also provide a powerful tool to investigate complex fuzzy set. Finally, an example application of signal detection demonstrates the utility of the (r, θ)-cut sets of complex fuzzy sets in practice.