Intuitionistic Fuzzy Mandelbrot Set

In recent years, the study of fuzzy sets has expanded to include various generalizations, such as intuitionistic fuzzy sets, which provide a richer framework for handling uncertainty. Despite significant progress, there remains a gap in understanding how these generalizations impact well-known mathematical constructs like the Mandelbrot set. This paper aims to introduce the intuitionistic fuzzy Mandelbrot set by delineating its membership function which is a measure of the degrees of inclusion of points to the set and its non-membership function, which quantifies the degrees of exclusion from the set under iteration, even for points with unbounded orbits.Furthermore, we delve into the exploration of the intuitionistic fuzzy Mandelbrot set’s basic topological properties, including the continuity of its membership and non-membership functions, as well as the symmetry exhibited by the set. Additionally, we examine its strong α,β-cut properties, such as its closedness, openness, boundedness, connectedness, and simply connectedness. We establish several key theorems, including the non-emptiness and boundedness of the fuzzy boundary of the Intuitionistic Fuzzy Mandelbrot Set (IFMS), although it is not necessarily compact. We also discuss applications in the field of image processing, such as edge detection, texture analysis, image segmentation, and pattern recognition.