Operation properties and δ-equalities of complex fuzzy sets

  • Authors:

  • Guangquan Zhang;Tharam Singh Dillon;Kai-Yuan Cai;Jun Ma;Jie Lu

  • Affiliations:

  • Faculty of Engineering and Information Technology, University of Technology, Sydney (UTS), P.O. Box 123, Broadway, NSW 2007, Australia;Curtin Business School, Curtin University of Technology, GPO Box U1987, WA 6845, Australia;Department of Automatic Control, Beijing University of Aeronautics and Astronautics, Beijing 100083, China;Faculty of Engineering and Information Technology, University of Technology, Sydney (UTS), P.O. Box 123, Broadway, NSW 2007, Australia;Faculty of Engineering and Information Technology, University of Technology, Sydney (UTS), P.O. Box 123, Broadway, NSW 2007, Australia

  • Venue:
  • International Journal of Approximate Reasoning
  • Year:
  • 2009

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Abstract

A complex fuzzy set is a fuzzy set whose membership function takes values in the unit circle in the complex plane. This paper investigates various operation properties and proposes a distance measure for complex fuzzy sets. The distance of two complex fuzzy sets measures the difference between the grades of two complex fuzzy sets as well as that between the phases of the two complex fuzzy sets. This distance measure is then used to define @d-equalities of complex fuzzy sets which coincide with those of fuzzy sets already defined in the literature if complex fuzzy sets reduce to real-valued fuzzy sets. Two complex fuzzy sets are said to be @d-equal if the distance between them is less than 1-@d. This paper shows how various operations between complex fuzzy sets affect given @d-equalities of complex fuzzy sets. An example application of signal detection demonstrates the utility of the concept of @d-equalities of complex fuzzy sets in practice.