Decomposition theorems and extension principle for complex Fuzzy sets

Complex fuzzy set was originally proposed as a mathematical tool to deal with uncertainty by taking amplitude term and phase term memberships of an element of a universal set. In this article, we study (𝛼, 𝜃)-cut sets of the complex fuzzy sets and describe some related properties of them. Based on these (𝛼, 𝜃)-cut sets, some decomposition theorems of the complex fuzzy sets are proposed. Moreover, the concept of Zadeh’s extension principle of the fuzzy sets is extended to the complex fuzzy sets and explored various related properties. Finally, some arithmetic operations are demonstrated for the complex fuzzy set by using the extension principle of the complex fuzzy sets. Numerical illustrations for each arithmetic operation are also given.