Provides detailed mathematical exposition of the fundamentals of fuzzy set theory, including intuitionistic fuzzy sets

This book examines fuzzy and intuitionistic fuzzy mathematics and unifies the latest existing works in literature. It enables readers to fully understand the mathematics of both fuzzy set and intuitionistic fuzzy set so that they can use either one in their applications.

Each chapter of Fuzzy Set and Its Extension: The Intuitionistic Fuzzy Set begins with an introduction, theory, and several examples to guide readers along. The first one starts by laying the groundwork of fuzzy/intuitionistic fuzzy sets, fuzzy hedges, and fuzzy relations. The next covers fuzzy numbers and explains Zadeh's extension principle. Then comes chapters looking at fuzzy operators; fuzzy similarity measures and measures of fuzziness; and fuzzy/intuitionistic fuzzy measures and fuzzy integrals. The book also: discusses the definition and properties of fuzzy measures; examines matrices and determinants of a fuzzy matrix; and teaches about fuzzy linear equations. Readers will also learn about fuzzy subgroups. The second to last chapter examines the application of fuzzy and intuitionistic fuzzy mathematics in image enhancement, segmentation, and retrieval. Finally, the book concludes with coverage the extension of fuzzy sets. This book:

• Covers both fuzzy and intuitionistic fuzzy sets and includes examples and practical applications
• Discusses intuitionistic fuzzy integrals and recent aggregation operators using Choquet integral, with examples
• Includes a chapter on applications in image processing using fuzzy and intuitionistic fuzzy sets
• Explains fuzzy matrix operations and features examples

Fuzzy Set and Its Extension: The Intuitionistic Fuzzy Set is an ideal text for graduate and research students, as well as professionals, in image processing, decision-making, pattern recognition, and control system design.

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.