On type-2 and type-3 Fuzzy sets

Abstract

This thesis presents a comprehensive mathematical review of the hierarchical evolution of fuzzy sets across their three generations, aiming to simplify the computational complexity and algebraic intractability of three-dimensional systems to make them practically computable. The first chapter addresses fuzzy relations and algebraic triangular norms (t-norms) for Type-1 systems, while the second chapter focuses on the geometric modeling of Interval Type-2 membership functions and the analysis of the Footprint of Uncertainty (FOU). Finally, the research concludes in the third chapter by investigating general Type-3 fuzzy hierarchies and employing alpha-plane (α-planes) decompositions to transform complex volumetric structures into crisp, actionable intervals, establishing these advanced systems as a vital cornerstone for next-generation artificial intelligence, intelligent control, and automated medical diagnosis.