Complex fuzzy sets

Daniel Ramot, Ron Milo, Menahem Friedman, Abraham Kandel

Research output: Contribution to journalArticlepeer-review

395 Scopus citations


The objective of this paper is to investigate the innovative concept of complex fuzzy sets. The novelty of the complex fuzzy set lies in the range of values its membership function may attain. In contrast to a traditional fuzzy membership function, this range is not limited to [0, 1], but extended to the unit circle in the complex plane. Thus, the complex fuzzy set provides a mathematical framework for describing membership in a set in terms of a complex number. The inherent difficulty in acquiring intuition for the concept of complex-valued membership presents a significant obstacle to the realization of its full potential. Consequently, a major part of this work is dedicated to a discussion of the intuitive interpretation of complex-valued grades of membership. Examples of possible applications, which demonstrate the new concept, include a complex fuzzy representation of solar activity (via measurements of the sunspot number), and a signal processing application. A comprehensive study of the mathematical properties of the complex fuzzy set is presented. Basic set theoretic operations on complex fuzzy sets, such as complex fuzzy complement, union, and intersection, are discussed at length. Two novel operations, namely set rotation and set reflection, are introduced. Complex fuzzy relations are also considered.

Original languageEnglish
Pages (from-to)171-186
Number of pages16
JournalIEEE Transactions on Fuzzy Systems
Issue number2
StatePublished - 1 Apr 2002
Externally publishedYes


  • Complex fuzzy intersection
  • Complex fuzzy relations
  • Complex fuzzy sets
  • Complex fuzzy union
  • Complex-valued grades of membership
  • Fuzzy complex numbers


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