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More About This Title Fuzzy Set and Its Extension - The IntuitionisticFuzzy Set
English
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.
English
Tamalika Chaira, PhD, is an advisory board member at Aravalli Pharma Health Care, New Delhi. She was a scientist for seven years in Indian Institute of Technology Delhi, New Delhi. Dr. Chaira received her PhD from Indian Institute of Technology Kharagpur, India. She is on the Advisory board of Advances in Information Mining. Her research is based on medical image processing using fuzzy, intuitionistic fuzzy, and Type II fuzzy mathematics.
English
Organization of the book
Preface
Chapter 1: Fuzzy/intuitionistic fuzzy set theory
1.1 Introduction to fuzzy set
1.2 Mathematical representation of fuzzy set
1.3 Membership function
1.4 Fuzzy relations
1.5 Projection
1.6 Composition of fuzzy relation
1.7 Fuzzy binary relation
1.8 Transitive closure of fuzzy binary relation
1.9 Fuzzy equivalence relation
1.10 Intuitionistic fuzzy set
1.11 Construction of intuitionistic fuzzy set
1.12 Intuitionistic fuzzy relations (IFR)
1.13 Composition of intuitionistic fuzzy relation
1.13.1 Composition of IFR using t norms and co norms
1.14 Intuitionistic fuzzy binary relation
1.15 Summary
1.16 References
Chapter 2: Fuzzy/intuitionistic fuzzy numbers
2.1 Introduction
2.2 Fuzzy numbers
2.3 Fuzzy intervals
2.4 Zadeh’s extension principle
2.5 Fuzzy numbers with α- intervals
2.6 Operation on fuzzy numbers with intervals
2.7 Operation on fuzzy numbers based on α- level
2.8 Operations on fuzzy numbers using extension principle
2.8.1 Examples on arithmetic operation on fuzzy numbers using extension principle
2.9 L-R representation of fuzzy numbers
2.10 Intuitionistic fuzzy number
2.11 Triangular intuitionistic fuzzy numbers
2.12 Operations using triangular intuitionistic fuzzy numbers
2.13 Trapezoidal intuitionistic fuzzy number
2.14 Cut set of triangular fuzzy number
2.15 Distance between two intuitionistic fuzzy number
2.16 Summary
2.17 References
Chapter 3: Similarity measures and measures of fuzziness
3.1 Introduction
3.2 Distance and similarity measures
3.3 Types of distance measure between fuzzy sets
3.4 Types of Similarity measures between fuzzy sets
3.5 Generalized fuzzy number
3.6 Similarity measures using fuzzy numbers
3.7 Inclusion measure
3.8 Measures of fuzziness
3.8.1 Index of fuzziness
3.8.2 Yager’s measure
3.8.3 Entropy
3.9 Intuitionistic fuzzy distance measures
3.10 Intuitionistic fuzzy entropy
3.11 Different types of entropies
3.12 Summary
References
Chapter 4: Fuzzy/Intuitionistic fuzzy measures and fuzzy integrals
4.1 Introduction
4.2 Definition of Fuzzy measure
4.3 Fuzzy measures
4.3.1 Sugeno λ- measure
4.3.2 Belief measure
4.3.3 Plausibility measure
4.3.4 Possibility and necessity measures Fuzzy integrals
4.4 Fuzzy integral
4.4.1 Sugeno integral
4.4.2 Choquet integral
4.4.3 Sipos Integral
4.5 Intuitionistic fuzzy Choquet integral
4.6 Summary
References
Chapter 5: Operations on fuzzy/intuitionistic fuzzy sets and application in decision making
5.1 Introduction
5.2 Fuzzy operation
5.3 Fuzzy other operators - t norms and t-conorms
5.4 Implication operator
5.5 Fuzzy aggregation operator with application in decision making
a) Fuzzy weighted averaging operator
b) Fuzzy ordered weighted averaging operator
c) Fuzzy generalized ordered weighted averaging operator (GOWA)
d) Fuzzy hybrid averaging operator(FHA)
e) Fuzzy quasi arithmetic weighted averaging operator
f) Induced fuzzy generalized averaging operator
g) Choquet aggregation operator
h) Induced Choquet ordered aggregation operator
5.6 Intuitionistic fuzzy operator
5.7 Intuitionistic fuzzy aggregation operator
a) Generalized intuitionistic fuzzy aggregation operator
b) Generalized intuitionistic fuzzy ordered weighting operator
c) Generalized intuitionistic fuzzy hybrid operator(FHA)
d) Intuitionistic fuzzy weighted geometric operator
e) Intuitionistic fuzzy ordered weighted geometric operator
f) Induced generalized intuitionistic fuzzy ordered averaging operator
g) Intuitionistic fuzzy Choquet integral operator
h) Induced intuitionistic fuzzy Choquet integral operator
5.8 Example on decision making problem
5.9 Summary
References
Chapter 6: Fuzzy linear equation
6.1 Introduction
6.2 Fuzzy linear equation
6.3 Solving linear equation using Cramer’s Rule
6.4 Inverse of a fuzzy matrix
6.5 Summary
References
Chapter 7: Fuzzy matrices and determinants
7.1 Basic matrix theory
7.2 Fuzzy matrices
7.3 Determinant of a square fuzzy matrix
7.3.1 Examples of fuzzy determinants
7.4 Adjoint of a square fuzzy matrix
7.4.1 Few proposition of adjoint fuzzy matrices
7.5 Reflexive fuzzy matrix
7.6 Generalized inverse of a fuzzy matrix
7.7 Intuitionistic fuzzy matrix
7.8 Summary
References
Chapter 8: Fuzzy subgroups
8.1 Introduction
8.2 Properties of fuzzy subgroups
8.3 Fuzzy level sub group
8.4 Fuzzy normal subgroup
8.5 Fuzzy subgroups using t-norms
8.6 Product of fuzzy subgroup
8.7 Summary
References
Chapter 9: Application of Fuzzy/Intuitionistic fuzzy set in image processing
9.1 Introduction
9.2 Digital images
9.3 Image enhancement
9.3.1 Fuzzy enhancement method
9.3.2 Intuitionistic fuzzy enhancement method
9.4 Image thresholding
9.4.1 Intuitionistic fuzzy thresholding method
9.4.2 Fuzzy thresholding method
9.5 Image edge detection
9.5.1 Fuzzy edge detection
9.5.2 Intuitionistic fuzzy detection
9.6 Clustering
9.6.1 Fuzzy c means clustering
9.6.2 Intuitionistic fuzzy clustering
9.6.3 Kernel clustering
9.7 Mathematical morphology
9.7.1 Fuzzy method
9.7.2 Intuitionistic fuzzy method
9.8 Summary
References
Chapter 10: Type 2 fuzzy set
10.1 Introduction
10.2 Type 2 fuzzy set
10.3 Operations on type 2 fuzzy set
10.4 Inclusion measure and similarity measure
10.5 Interval type 2 fuzzy set
10.6 Application of interval type 2 fuzzy set in image processing
10.7 Summary
References
Problems
