Mathematical Foundations of Fuzzy Sets

Mathematical Foundations of Fuzzy Sets

by Hsien-Chung Wu
Released January 2023
Publisher(s): Wiley
ISBN: 9781119981527

Read it now on the O’Reilly learning platform with a 10-day free trial.

O’Reilly members get unlimited access to books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.

Start your free trial

Book description

Mathematical Foundations of Fuzzy Sets

Introduce yourself to the foundations of fuzzy logic with this easy-to-use guide

Many fields studied are defined by imprecise information or high degrees of uncertainty. When this uncertainty derives from randomness, traditional probabilistic statistical methods are adequate to address it; more everyday forms of vagueness and imprecision, however, require the toolkit associated with 'fuzzy sets' and 'fuzzy logic'. Engineering and mathematical fields related to artificial intelligence, operations research and decision theory are now strongly driven by fuzzy set theory.

Mathematical Foundations of Fuzzy Sets introduces readers to the theoretical background and practical techniques required to apply fuzzy logic to engineering and mathematical problems. It introduces the mathematical foundations of fuzzy sets as well as the current cutting edge of fuzzy-set operations and arithmetic, offering a rounded introduction to this essential field of applied mathematics. The result can be used either as a textbook or as an invaluable reference for working researchers and professionals.

Mathematical Foundations of Fuzzy Sets offers thereader:

  • Detailed coverage of set operations, fuzzification of crisp operations, and more
  • Logical structure in which each chapter builds carefully on previous results
  • Intuitive structure, divided into 'basic' and 'advanced' sections, to facilitate use in one- or two-semester courses

Mathematical Foundations of Fuzzy Sets is essential for graduate students and academics in engineering and applied mathematics, particularly those doing work in artificial intelligence, decision theory, operations research, and related fields.

Table of contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Preface
  5. 1 Mathematical Analysis
    1. 1.1 Infimum and Supremum
    2. 1.2 Limit Inferior and Limit Superior
    3. 1.3 Semi‐Continuity
    4. 1.4 Miscellaneous
  6. 2 Fuzzy Sets
    1. 2.1 Membership Functions
    2. 2.2 ‐level Sets
    3. 2.3 Types of Fuzzy Sets
  7. 3 Set Operations of Fuzzy Sets
    1. 3.1 Complement of Fuzzy Sets
    2. 3.2 Intersection of Fuzzy Sets
    3. 3.3 Union of Fuzzy Sets
    4. 3.4 Inductive and Direct Definitions
    5. 3.5 ‐Level Sets of Intersection and Union
    6. 3.6 Mixed Set Operations
  8. 4 Generalized Extension Principle
    1. 4.1 Extension Principle Based on the Euclidean Space
    2. 4.2 Extension Principle Based on the Product Spaces
    3. 4.3 Extension Principle Based on the Triangular Norms
    4. 4.4 Generalized Extension Principle
  9. 5 Generating Fuzzy Sets
    1. 5.1 Families of Sets
    2. 5.2 Nested Families
    3. 5.3 Generating Fuzzy Sets from Nested Families
    4. 5.4 Generating Fuzzy Sets Based on the Expression in the Decomposition Theorem
    5. 5.5 Generating Fuzzy Intervals
    6. 5.6 Uniqueness of Construction
  10. 6 Fuzzification of Crisp Functions
    1. 6.1 Fuzzification Using the Extension Principle
    2. 6.2 Fuzzification Using the Expression in the Decomposition Theorem
    3. 6.3 The Relationships between EP and DT
    4. 6.4 Differentiation of Fuzzy Functions
    5. 6.5 Integrals of Fuzzy Functions
  11. 7 Arithmetics of Fuzzy Sets
    1. 7.1 Arithmetics of Fuzzy Sets in
    2. 7.2 Arithmetics of Fuzzy Vectors
    3. 7.3 Difference of Vectors of Fuzzy Intervals
    4. 7.4 Addition of Vectors of Fuzzy Intervals
    5. 7.5 Arithmetic Operations Using Compatibility and Associativity
    6. 7.6 Binary Operations
    7. 7.7 Hausdorff Differences
    8. 7.8 Applications and Conclusions
  12. 8 Inner Product of Fuzzy Vectors
    1. 8.1 The First Type of Inner Product
    2. 8.2 The Second Type of Inner Product
  13. 9 Gradual Elements and Gradual Sets
    1. 9.1 Gradual Elements and Gradual Sets
    2. 9.2 Fuzzification Using Gradual Numbers
    3. 9.3 Elements and Subsets of Fuzzy Intervals
    4. 9.4 Set Operations Using Gradual Elements
    5. 9.5 Arithmetics Using Gradual Numbers
  14. 10 Duality in Fuzzy Sets
    1. 10.1 Lower and Upper Level Sets
    2. 10.2 Dual Fuzzy Sets
    3. 10.3 Dual Extension Principle
    4. 10.4 Dual Arithmetics of Fuzzy Sets
    5. 10.5 Representation Theorem for Dual‐Fuzzified Function
  15. Bibliography
  16. Mathematical Notations
  17. Index
  18. End User License Agreement

Product information

  • Title: Mathematical Foundations of Fuzzy Sets
  • Author(s): Hsien-Chung Wu
  • Release date: January 2023
  • Publisher(s): Wiley
  • ISBN: 9781119981527