A New Interpretation of Complex Membership Grade

Complex fuzzy sets utilize a complex degree of membership, represented in polar coordinates, which is a combination of a degree of membership in a fuzzy set along with a crisp phase value that denotes position within the set. The compound value carries more information than a traditional fuzzy set and enables efficient reasoning. In this paper, we present a new and generalized interpretation of a complex grade of membership, where a complex membership grade defines a complex fuzzy class. The new definition provides rich semantics that is not readily available through traditional fuzzy sets or complex fuzzy sets and is not limited to a compound of crisp cyclical data with fuzzy data. Furthermore, the two components of the complex fuzzy class carry fuzzy information. A complex class is represented either in Cartesian or in polar coordinates where both axes induce fuzzy interpretation. Another novelty of the scheme is that it enables representing an infinite set of fuzzy sets. The paper provides the new definition of complex fuzzy classes along with axiomatic definition of basic operations on complex fuzzy classes. In addition, coordinate transformation as well as an extension from two-dimensional fuzzy classes to n-dimensional fuzzy classes are presented. (C) 2011 Wiley Periodicals, Inc.