Fuzzy Set and Its Extension: The Intuitionistic Fuzzy Set

This book examines fuzzy and intuitionistic fuzzy mathematics and unifies the latest existing works in literature. The book contains 10 chapters. Each chapter begins with an introduction, theory, and several examples to guide readers. Chapter 1 begins with the groundwork of fuzzy/intuitionistic fuzzy sets, fuzzy hedges, and fuzzy relations. Different types of membership function with examples, terms related to membership function, are explained. Chapter 2 covers fuzzy numbers. Zadeh’s extension principle is explained, which states how an image of a fuzzy subset is formed using a function. Next, Chapter 3 discusses fuzzy operators where different types of fuzzy operators are used. Fuzzy algebraic operations such as complement, sum, difference, bounded sum, bounded difference and others are explained, with examples. Chapter 4 details fuzzy similarity measures and measures of fuzziness. Similarity measures on fuzzy sets and fuzzy numbers are discussed. Chapter 5 outlines fuzzy/intuitionistic fuzzy measures and fuzzy integrals. Definition and properties of fuzzy measures are discussed. Then, Chapter 6 examines matrices and determinants of a fuzzy matrix. Fuzzy matrix operations are explained with examples. Fuzzy linear equations are covered in Chapter 7. Chapter 8 is dedicated to fuzzy subgroups. The definition of a fuzzy subgroup is provided along with its properties. Many examples of fuzzy subgroups are provided. Chapter 9 examines application of fuzzy and intuitionistic fuzzy mathematics in image enhancement, segmentation, retrieval. Finally, the book concludes with Chapter 10, which covers the extension of fuzzy sets.