From ambiguous sets to single-valued ambiguous complex numbers: Applications in Mandelbrot set generation and vector directions

Abstract

The concept of an ambiguous set (AS) was initially introduced by Singh et al. [P. Singh, Y.-P. Huang and T.-T. Lee — A novel ambiguous set theory to represent uncertainty and its application to brain MR image segmentation, in Proc. IEEE Int. Conf. Systems, Man and Cybernetics (SMC), Bari, Italy, 2019, pp. 2460–2465]. Can we examine the possibility of expanding it to introduce ambiguous complex numbers? This study aims to answer this question by first introducing the concept of an AS and then providing a set of formulas that aid in determining the membership degrees of the set. Then, this study expands the idea of AS to a single-valued ambiguous number (SVAN). Based on SVAN, the concept of single-valued ambiguous complex number (SVACN) is illustrated. Following that, various theories and properties of SVACN are addressed in terms of their complex conjugate, polar, exponential forms, and roots. Furthermore, the study proposes an algorithm tailored to generating the Mandelbrot set using varying numbers of SVACNs. Additionally, this approach furnishes directional vectors for the SVACNs within a two-dimensional plane. The study also deliberates on the manifold advantages of this methodology, particularly emphasizing its capacity to provide directional guidance for SVACNs.