Complex fuzzy sets: towards new foundations

Uncertainty of complex-valued physical quantities z=x+y can be described by complex fuzzy sets. Such sets can be described by membership functions μ(x, y) which map the universe of discourse (complex plane) into the interval [0, 1]. The problem with this description is that it is difficult to directly translate into words from natural language. To make this translation easier, several authors have proposed to use, instead of a single membership function for describing the complex number, several membership functions which describe different real-valued characteristics of this number, such as its real part, its imaginary part, its absolute value, etc. The quality of this new description strongly depends on the choice of these real-valued functions, so it is important to choose them optimally. We formulate the problem of optimal choice of these functions and show that, for all reasonable optimality criteria, the level sets of optimal functions are straight lines and circles. This theoretical result is in good accordance with our numerical experiments, according to which such functions indeed lead to a good description of complex fuzzy sets