Operation properties and (α,β)-equalities of complex intuitionistic fuzzy sets

Complex intuitionistic fuzzy set, as an extension of intuitionistic fuzzy set, 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. In this paper, various operation properties of complex intuitionistic fuzzy sets are investigated when the membership phase and non-membership phase are restricted to [ 0 , 2 π ] . Meanwhile, consider that precise membership values and non-membership values should normally be of no practical significance, and there is no equality and proximity measure investigation on complex intuitionistic fuzzy sets. First of all, we proposed a new distance measure for complex intuitionistic fuzzy sets. Then this distance measure is used to define ( α , β ) -equalities of complex intuitionistic fuzzy sets. Two complex intuitionistic fuzzy sets are said to be ( α , β ) -equal if the distance between their membership degrees is less than 1 - α and the distance between their non-membership degrees is less than β . Furthermore, this paper shows how various operations between complex intuitionistic fuzzy sets affect given ( α , β ) -equalities of complex intuitionistic fuzzy sets. Finally, complex intuitionistic fuzzy relations are discussed and some examples are given to illuminate the results obtained in this paper.