Delta-equalities of Complex Fuzzy Relations

A complex fuzzy relation is defined as a fuzzy relation whose membership function takes values in the unit circle on a complex plane. This paper first investigates various operation properties of a complex fuzzy relation. It then defines the distance measure of two complex fuzzy relations that can measure the differences between the grades as well as the phases of two complex fuzzy relations. This distance measure is used to define δ-equalities of complex fuzzy relations that coincide with those of fuzzy relations already defined in the literature if complex fuzzy relations reduce to real-valued fuzzy relations. Two complex fuzzy relations are said to be δ-equal if the distance between them is less than 1-δ. This paper shows how various operations between complex fuzzy relations, including T-norms and S-norms, affect given δ-equalities of complex fuzzy relations. Finally, fuzzy inference is examined in the framework of delta- equalities of complex fuzzy relations.