Complex Number Representation of Intuitionistic Fuzzy Sets Dan E. Tamir Department of Computer Science Texas State University San Marcos, Texas USA dt19@txstate.edu Mumtaz Ali Department of Mathematics Quaid-i-azam University Islamabad 45320, Pakistan mumtazali7288@gmail.com Naphtali D. Rishe, and Abraham Kandel School of Computing and Information Sciences Florida International University Miami, Florida, USA rishen@fiu.edu, abraham.kandel.fiu@gmail.com Abstract – Complex numbers can capture compound features and convey multifaceted information; thereby providing means for solving complicated problems. In this paper, we combine the degree of membership and a degree of non-membership of members of Intuitionistic Fuzzy Sets via complex numbers to characterize these fuzzy sets. This approach enables extending several concepts such as classical fuzzy sets, Pythagorean fuzzy sets, and complex fuzzy sets. We discuss complex numbers-based set theoretic operations such as union, intersection, and complement. We define the No-Man-Zone (NMZ) set and establish the relation of NMZ characterization with complex numbers. Further, we introduce athematic complex numbers-based operations of intuitionistic fuzzy sets. We show that the square of the absolute of an intuitionistic fuzzy set becomes a Pythagorean fuzzy set. The polar form of an intuitionistic fuzzy set is reduced to a complex fuzzy set. Keywords: Fuzzy Set Theory, Intuitionistic Fuzzy Set Theory I. INTRODUCTION Fuzzy sets and fuzzy logic were introduced by Zadeh in 1965 in order to handle uncertainty and ambiguity [42, 43]. A fuzzy set is characterized by a membership degree whose range is the unit interval. Fuzzy logic is a multilevel extension to the classical logic such that proposition can get any value in the unit interval instead of one of the two values ‘True’ or ‘False’. Based on the theory of fuzzy sets, several additional concepts, such as interval valued fuzzy sets [43], type-2 fuzzy sets [27, 15, 43], and intuitionistic fuzzy sets [4, 5], have been developed in order to effectively handle uncertainty. Fuzzy sets and fuzzy logic have applications in signal processing [23, 27, 28], control theory [19, 40, 44,], reasoning [22], and data mining [16]. Additional background on fuzzy sets can be found in [6, 7, 13, 31, 35, 36, 37, 46, 47]. Intuitionistic fuzzy sets were introduced by Atanassove in 1986 as a generalization of fuzzy sets by adding the degree of non-membership into the fuzzy set [4]. Thus, an intuitionistic fuzzy set is characterized by a degree of membership (say ) and a degree of non-membership (). And the sum of  and  is restricted to be  + ≤ 1. Philosophically this is a bit problematic since we generally assume that the degree of membership and the degree of non-membership of an element of a fuzzy set sums up to 1. The problem is that the residual (1 − (  + )) is implying the existence of another fuzzy set. This fuzzy set is called the hesitant zone. We refer to this set as the no man zone (NMZ). A good example for real life occurrence of intuitionistic fuzzy sets is the case of getting advice from two experts. For example, consider a person that consults with two stock brokers. One expert might advise for “strong buy” on a specific stock and the second might advise for “strong sell.” Both terms are implying membership (and non-membership). The combination constructs an intuitionistic fuzzy set along with a zone of uncertainty or hesitation. Intuitionistic fuzzy sets often better represent fuzziness. Intuitionistic fuzzy sets have been successfully applied in the fields of modeling imprecision [17], decision making problems [27], pattern recognition [40], economics [20], computational intelligence [14], and medical diagnosis [34]. The strength of these concepts evolves from cases where conflicting information concerning membership taints the ability to determine the actual fuzzy membership of objects. Complex fuzzy set and logic, which are extensions of fuzzy sets and logic respectively, were first proposed by Ramot et al. [32, 33]. According to their definition, a complex fuzzy set is characterized by a complex grade of membership, which is a combination of a traditional fuzzy degree of membership,