Operation properties and delta-equalities of complex fuzzy classes

A complex fuzzy class is a set of fuzzy sets which is characterized by a pure complex fuzzy grade of membership where both the real and imaginary parts are fuzzy functions. The values that a pure complex fuzzy grade of membership may receive all lie within the unite square or unit circle in the complex plane. In this paper, we investigate different operation properties and propose a distance measure for complex fuzzy classes. The distance of two complex fuzzy classes measures the difference between the memberships of the fuzzy sets in the two complex fuzzy classes as well as the difference between the memberships in the related fuzzy sets in the two complex fuzzy classes. d-equalities of two complex fuzzy classes are then defined which mainly base on this distance measure. If the distance between two complex fuzzy classes is less than or equal to d, then they are said to be d-equal. This paper reveals that different operations between complex fuzzy classes can affect given delta-equalities of complex fuzzy classes. Further, an application of utilizing the concept of d-equalities of complex fuzzy classes in stocks and mutual funds in the stock market is presented.