This edition first published 2019
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Since Lofti A. Zadeh introduced fuzzy set theory about 50 years ago, i.e. in 1965, theory of fuzzy sets has evolved in many directions and has received more attention from many researchers. Applications of the theory can be found ranging from pattern recognition, control system, image processing, decision making, operations research, robotics, and management.
This book discusses on connections between fuzzy set and crisp set, fuzzy relations, operations on fuzzy sets, various aggregation operators using fuzzy sets, fuzzy numbers, arithmetic operations on fuzzy numbers, fuzzy integrals, fuzzy matrices and determinants, and fuzzy groups. Applications on decision making and image processing is also given.
Apart from fuzzy set, intuitionistic fuzzy set is also discussed in this book. Since its inception by K. Atanassov in 1985, intuitionistic fuzzy set theory has also received attention but to limited number of researchers as compared to fuzzy set. Though its use in application is not as comparable as that of fuzzy set, but still research studies are carried out in the areas that use fuzzy set. In intuitionistic fuzzy set, computational complexity is more as two types of uncertainties are used. But, for obtaining better result, where uncertainty present is more, especially in diagnosis of medical images, accurate result is very much important compromising the computational complexity. So, researchers try to use it on real‐time application.
The book discusses the basics of intuitionistic fuzzy set, intuitionistic fuzzy relations, operations on intuitionistic fuzzy sets, various intuitionistic fuzzy aggregation operators, intuitionistic fuzzy numbers, arithmetic operations on intuitionistic fuzzy numbers, intuitionistic fuzzy integrals, and intuitionistic fuzzy matrices. Also, application in decision making and image processing using intuitionistic fuzzy set is also given.
This book is an attempt to unify both fuzzy/intuitionistic fuzzy set and their existing work in application. The primary goal of this book is to help the readers to know the mathematics of both fuzzy set and intuitionistic fuzzy set so that with both these concepts, they can use either fuzzy/intuitionistic fuzzy set in their applications.
Finally, I would like to acknowledge the authors of the papers that have been referred in the book. I acknowledge my beloved daughter, Shruti De, for giving the title of the book. I acknowledge my parents for their continuous support while writing this book. I am also indebted to John Wiley & Sons, Inc. for making the publication of this book possible.
The book contains 10 chapters. Each chapter begins with an introduction, theory, and also several examples that will help the readers to understand the chapters in a better way. Chapter 1 starts with preliminaries of fuzzy sets and relations. Different types of membership function, composition of fuzzy relation, and fuzzy binary relation that includes symmetric, reflexive, transitive, and equivalent relations are explained with examples. Similar to fuzzy set, intuitionistic fuzzy sets, operations, relations, and compositions are also explained with examples.
Chapter 2 deals with fuzzy numbers. Zadeh’s extension principle is explained that states how an image of a fuzzy subset is formed using a function. Using the extension principle, arithmetic operations on fuzzy numbers are explained. Fuzzy numbers with α‐cut, operations on fuzzy numbers, and L–R representation of fuzzy numbers are explained with examples. Intuitionistic fuzzy numbers such as triangular and trapezoidal fuzzy numbers, along with operations, are also explained with examples.
Chapter 3 details fuzzy similarity measures and measures of fuzziness. Similarity measures on fuzzy sets and fuzzy numbers are discussed. More emphasis is given on similarity measure on fuzzy numbers. Different types of similarity measures based on the center of gravity, area, perimeter, and graded mean integration of fuzzy numbers are discussed in detail. Measures of fuzziness and different types of entropy are also explained. Intuitionistic fuzzy similarity measures, distance measures, and entropy are also discussed.
Chapter 4 outlines fuzzy measures and fuzzy integrals. Definition and properties of fuzzy measures are discussed. Sugeno measure is a special type of fuzzy measure is discussed with examples. Other types of fuzzy measures such as belief measure, possibility measure, plausibility measure, and necessity measure are discussed. Fuzzy integrals such as Choquet and Sugeno integrals are explained with figures and example on decision making problem is also provided. Intuitionistic fuzzy Choquet integral is also discussed.
Chapter 5 discusses on fuzzy operators where different types of fuzzy operators are used. Fuzzy algebraic operations such as complement, sum, difference, bounded sum, bounded difference, union, and intersection are explained with examples. Fuzzy set theoretic operations that include fuzzy triangular norms (t‐norms) and triangular conorms (t‐conorms) are explained. Triangular norms suggested by different authors are discussed. Fuzzy/intuitionistic fuzzy aggregation operators that combine different pieces of information into a single object in a same set are explained. Different types of aggregation operators such as fuzzy/intuitionistic fuzzy generalized ordered weighted averaging, hybrid averaging operator, quasi‐arithmetic weighted averaging operator, fuzzy/intuitionistic fuzzy‐induced generalized averaging operator, fuzzy/intuitionistic fuzzy Choquet and induced Choquet ordered aggregation operator are explained with examples on decision making.
Chapter 6 examines matrices and determinants of a fuzzy matrix. Fuzzy matrix/determinant operations are explained with properties and examples. Adjoint and determinants of a fuzzy matrix, and inverse of a fuzzy matrix are discussed with examples. Intuitionistic fuzzy determinants and matrices are also discussed.
Chapter 7 outlines fuzzy linear equation. It is a continuation of Chapter 6 where an unknown vector is computed using general equation method and also using Cramer’s rule. Finding inverse of a fuzzy matrix is discussed with examples. Fuzzy linear equation using L–R‐type fuzzy numbers is also discussed with examples where left and right spread of a fuzzy number are considered.
Chapter 8 is dedicated to fuzzy subgroups. Definition of fuzzy subgroup is provided along with properties. Many examples of fuzzy subgroups are mentioned. Other types of fuzzy subgroups such as fuzzy‐level subgroup and fuzzy normal subgroup are also discussed with examples. Definition of fuzzy subgroup with respect to fuzzy t‐norm is also included. Product of fuzzy subgroups with respect to t‐norm with propositions are explained.
Chapter 9 is based on the application on image processing. Introduction on image processing along with image enhancement, segmentation, clustering, edge detection, and morphology are explained with examples using both fuzzy and intuitionistic fuzzy set. Results on medical images for detection of abnormal lesions/clot/hemorrhage in CT scan brain image are shown.
Lastly, the book ends up with Chapter 10 where Type‐2 fuzzy set is explained. Introduction and representation of Type‐2 fuzzy set is discussed. Operations on Type‐2 fuzzy set along with examples are provided.