Fuzzy Set and its Extension. The Intuitionistic Fuzzy Set

Description

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. Contents......Page 6 Preface......Page 12 Organization of the Book......Page 14 1.1 Introduction to Fuzzy Set......Page 16 1.2 Mathematical Representation of Fuzzy Sets......Page 18 1.3 Membership Function......Page 21 1.4 Fuzzy Relations......Page 25 1.5 Projection......Page 28 1.6 Composition of Fuzzy Relation......Page 29 1.7 Fuzzy Binary Relation......Page 34 1.8 Transitive Closure of Fuzzy Binary Relation......Page 36 1.9 Fuzzy Equivalence Relation......Page 38 1.10 Intuitionistic Fuzzy Set......Page 39 1.11 Construction of Intuitionistic Fuzzy Set......Page 41 1.12 Intuitionistic Fuzzy Relations......Page 44 1.13 Composition of Intuitionistic Fuzzy Relation......Page 46 1.14 Intuitionistic Fuzzy Binary Relation......Page 49 1.15 Summary......Page 54 2.2 Fuzzy Numbers......Page 56 2.3 Fuzzy Intervals......Page 57 2.4 Zadeh's Extension Principle......Page 58 2.5 Fuzzy Numbers with α-Levels......Page 63 2.6 Operations on Fuzzy Numbers with Intervals......Page 67 2.7 Operations with Fuzzy Numbers based on α-Levels......Page 69 2.8 Operations on Fuzzy Numbers using Extension Principle......Page 77 2.9 L-R Representation of Fuzzy Numbers......Page 81 2.10 Intuitionistic Fuzzy Numbers......Page 88 2.11 Triangular Intuitionistic Fuzzy Number......Page 89 2.12 Operations Using Triangular Intuitionistic Fuzzy Numbers......Page 90 2.13 Trapezoidal Intuitionistic Fuzzy Numbers......Page 92 2.14 Cut Set of Intuitionistic Fuzzy Number......Page 93 2.16 Summary......Page 95 3.2 Distance and Similarity Measures......Page 97 3.3 Types of Distance Measure Between Fuzzy Sets......Page 98 3.5 Generalized Fuzzy Number......Page 99 3.6 Similarity Measures Between Two Fuzzy Numbers......Page 102 3.7 Inclusion Measure......Page 108 3.8 Measures of Fuzziness......Page 109 3.9 Intuitionistic FuzzyDistance and SimilarityMeasures......Page 112 3.10 Intuitionistic Fuzzy Entropy......Page 119 3.11 Different Types of Intuitionistic Fuzzy Entropies......Page 120 3.12 Summary......Page 121 4.2 Definition of Fuzzy Measure......Page 124 4.3 Fuzzy Measures......Page 125 4.4 Fuzzy Integrals......Page 134 4.5 Intuitionistic Fuzzy Integral......Page 143 4.6 Summary......Page 144 5.2 Fuzzy Operations......Page 146 5.3 Fuzzy Other Operators: Fuzzy T-Norms and T-Conorms......Page 151 5.4 Implication Operator......Page 155 5.5 Aggregation Operator with Application in Decision Making......Page 157 5.6 Intuitionistic Fuzzy Operators......Page 165 5.7 Intuitionistic Fuzzy Aggregation Operator......Page 166 5.8 Example on Decision-making Problems......Page 177 5.9 Summary......Page 181 6.1 Introduction......Page 184 6.2 Fuzzy Linear Equation......Page 185 6.3 Solving Linear Equation using Cramer's Rule......Page 190 6.4 Inverse of a Fuzzy Matrix......Page 195 6.5 Summary......Page 202 7.1 Basic Matrix Theory......Page 204 7.2 Fuzzy Matrices......Page 207 7.3 Determinant of Square Fuzzy Matrix......Page 215 7.4 Adjoint of Square Fuzzy Matrix......Page 219 7.5 Properties of Reflexive Matrices......Page 225 7.6 Generalized Inverse of a Fuzzy Matrix......Page 228 7.7 Intuitionistic Fuzzy Matrix......Page 229 7.8 Summary......Page 231 8.1 Introduction......Page 233 8.2 Theorems of Fuzzy Subgroup......Page 234 8.3 Fuzzy-level Subgroup......Page 238 8.4 Fuzzy Normal Subgroup......Page 240 8.5 Fuzzy Subgroups Using T-norms......Page 241 8.6 Product of Fuzzy Subgroups......Page 243 8.7 Summary......Page 246 9.2 Digital Images......Page 248 9.3 Image Enhancement......Page 249 9.4 Thresholding......Page 251 9.5 Edge Detection......Page 255 9.6 Clustering......Page 259 9.7 Mathematical Morphology......Page 263 9.8 Summary......Page 267 10.1 Introduction......Page 269 10.2 Type-2 Fuzzy Set......Page 270 10.3 Operations on Type-2 Fuzzy Set......Page 273 10.4 Inclusion Measure and Similarity Measure......Page 277 10.5 Interval Type-2 Fuzzy Set......Page 280 10.6 Application of Interval Type-2 Fuzzy Set in Image Segmentation......Page 281 10.7 Summary......Page 283 Beyond your Doubts......Page 284 Index......Page 290