Fuzzy Set and Its Extension

Fuzzy Set and Its Extension

by Tamalika Chaira
Released April 2019
Publisher(s): Wiley
ISBN: 9781119544197

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Book description

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. 

Table of contents

  1. Cover
  2. Preface
  3. Organization of the Book
  4. 1 Fuzzy/Intuitionistic Fuzzy Set Theory
    1. 1.1 Introduction to Fuzzy Set
    2. 1.2 Mathematical Representation of Fuzzy Sets
    3. 1.3 Membership Function
    4. 1.4 Fuzzy Relations
    5. 1.5 Projection
    6. 1.6 Composition of Fuzzy Relation
    7. 1.7 Fuzzy Binary Relation
    8. 1.8 Transitive Closure of Fuzzy Binary Relation
    9. 1.9 Fuzzy Equivalence Relation
    10. 1.10 Intuitionistic Fuzzy Set
    11. 1.11 Construction of Intuitionistic Fuzzy Set
    12. 1.12 Intuitionistic Fuzzy Relations
    13. 1.13 Composition of Intuitionistic Fuzzy Relation
    14. 1.14 Intuitionistic Fuzzy Binary Relation
    15. 1.15 Summary
    16. References
  5. 2 Playing with Fuzzy/Intuitionistic Fuzzy Numbers
    1. 2.1 Introduction
    2. 2.2 Fuzzy Numbers
    3. 2.3 Fuzzy Intervals
    4. 2.4 Zadeh’s Extension Principle
    5. 2.5 Fuzzy Numbers with α‐Levels
    6. 2.6 Operations on Fuzzy Numbers with Intervals
    7. 2.7 Operations with Fuzzy Numbers based on α‐Levels
    8. 2.8 Operations on Fuzzy Numbers Using Extension Principle
    9. 2.9 L–R Representation of Fuzzy Numbers
    10. 2.10 Intuitionistic Fuzzy Numbers
    11. 2.11 Triangular Intuitionistic Fuzzy Number
    12. 2.12 Operations Using Triangular Intuitionistic Fuzzy Numbers
    13. 2.13 Trapezoidal Intuitionistic Fuzzy Numbers
    14. 2.14 Cut Set of Intuitionistic Fuzzy Number
    15. 2.15 Distances Between Two Intuitionistic Fuzzy Numbers
    16. 2.16 Summary
    17. References
  6. 3 Similarity Measures and Measures of Fuzziness
    1. 3.1 Introduction
    2. 3.2 Distance and Similarity Measures
    3. 3.3 Types of Distance Measure Between Fuzzy Sets
    4. 3.4 Types of Similarity Measures Between Fuzzy Sets
    5. 3.5 Generalized Fuzzy Number
    6. 3.6 Similarity Measures Between Two Fuzzy Numbers
    7. 3.7 Inclusion Measure
    8. 3.8 Measures of Fuzziness
    9. 3.9 Intuitionistic Fuzzy Distance and Similarity Measures
    10. 3.10 Intuitionistic Fuzzy Entropy
    11. 3.11 Different Types of Intuitionistic Fuzzy Entropies
    12. 3.12 Summary
    13. References
  7. 4 Fuzzy/Intuitionistic Fuzzy Measures and Fuzzy Integrals
    1. 4.1 Introduction
    2. 4.2 Definition of Fuzzy Measure
    3. 4.3 Fuzzy Measures
    4. 4.4 Fuzzy Integrals
    5. 4.5 Intuitionistic Fuzzy Integral
    6. 4.6 Summary
    7. References
  8. 5 Operations on Fuzzy/Intuitionistic Fuzzy Sets and Application in Decision Making
    1. 5.1 Introduction
    2. 5.2 Fuzzy Operations
    3. 5.3 Fuzzy Other Operators: Fuzzy T‐Norms and T‐Conorms
    4. 5.4 Implication Operator
    5. 5.5 Aggregation Operator with Application in Decision Making
    6. 5.6 Intuitionistic Fuzzy Operators
    7. 5.7 Intuitionistic Fuzzy Aggregation Operator
    8. 5.8 Example on Decision‐making Problems
    9. 5.9 Summary
    10. References
  9. 6 Fuzzy Linear Equations
    1. 6.1 Introduction
    2. 6.2 Fuzzy Linear Equation
    3. 6.3 Solving Linear Equation Using Cramer's Rule
    4. 6.4 Inverse of a Fuzzy Matrix
    5. 6.5 Summary
    6. References
  10. 7 Fuzzy Matrices and Determinants
    1. 7.1 Basic Matrix Theory
    2. 7.2 Fuzzy Matrices
    3. 7.3 Determinant of a Square Fuzzy Matrix
    4. 7.4 Adjoint of a Square Fuzzy Matrix
    5. 7.5 Properties of Reflexive Matrices
    6. 7.6 Generalized Inverse of a Fuzzy Matrix
    7. 7.7 Intuitionistic Fuzzy Matrix
    8. 7.8 Summary
    9. References
  11. 8 Fuzzy Subgroups
    1. 8.1 Introduction
    2. 8.2 Theorems of Fuzzy Subgroup
    3. 8.3 Fuzzy‐level Subgroup
    4. 8.4 Fuzzy Normal Subgroup
    5. 8.5 Fuzzy Subgroups Using T‐norms
    6. 8.6 Product of Fuzzy Subgroups
    7. 8.7 Summary
    8. References
  12. 9 Application of Fuzzy/Intuitionistic Fuzzy Set in Image Processing
    1. 9.1 Introduction
    2. 9.2 Digital Images
    3. 9.3 Image Enhancement
    4. 9.4 Thresholding
    5. 9.5 Edge Detection
    6. 9.6 Clustering
    7. 9.7 Mathematical Morphology
    8. 9.8 Summary
    9. References
  13. Chapter 10: Type‐2 Fuzzy Set
    1. 10.1 Introduction
    2. 10.2 Type‐2 Fuzzy Set
    3. 10.3 Operations on Type‐2 Fuzzy Set
    4. 10.4 Inclusion Measure and Similarity Measure
    5. 10.5 Interval Type‐2 Fuzzy Set
    6. 10.6 Application of Interval Type‐2 Fuzzy Set in Image Segmentation
    7. 10.7 Summary
    8. References
  14. Beyond Your Doubts
  15. Index
  16. End User License Agreement

Product information

  • Title: Fuzzy Set and Its Extension
  • Author(s): Tamalika Chaira
  • Release date: April 2019
  • Publisher(s): Wiley
  • ISBN: 9781119544197