A Novel Approach Toward Complex Pythagorean Fuzzy Sets and Their Applications in Visualization Technology

The theory of complex Pythagorean fuzzy set (CPFS) has been already interpreted. This theory states that in both polar and Cartesian form the degree of membership and degree of non-membership are located in a complex plane’s unit disc. However the Cartesian presentation has some drawbacks for instance, this theory can’t consider the full belongingness
$\left ({{ 1+\iota 1, 0+\iota 0 }}\right)$
of the element because this value is not located in the unit disc of a complex plane but in a unit square of a complex plane. Hence, with the existing Cartesian form of CPFS, the full belongingness of the element can’t be described. Further, in polar form, the degrees of membership and non-membership are interpreted by amplitude and phase terms. Where the amplitude acts similar as they act in the Pythagorean fuzzy set (PFS) while phase terms show the periodicity, direction, or position of the element in a set. However, this description of degrees of membership and non-membership is confined to the polar structure and is inappropriate for inclusion in logical operations accompanied by CPFS in Cartesian coordinates. Therefore, in this article, we devise a theory of CPFS in Cartesian coordinates, where both degrees of membership and non-membership are in Cartesian form located in a complex plane’s unit disc and containing real and imaginary terms. These terms are fuzzy functions and carry fuzzy information. We also establish a few critical and basic operations for CPFS in Cartesian coordinate and then discuss some aggregation operators (AOs) and a multi-attribute decision-making (MADM) method within CPFS. After that, we employ the established theory in the field of visualization technology to reveal the applicability and requirement of the proposed theory. We interpreted the comparison of the deduced theory with certain prevailing theories to portray the supremacy of this work.