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A Modern Introduction to Fuzzy Mathematics

Release Date: 2020/07/01
ISBN: 9781119445289
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Provides readers with the foundations of fuzzy mathematics as well as more advanced topics

A Modern Introduction to Fuzzy Mathematics provides a concise presentation of fuzzy mathematics., moving from proofs of important results to more advanced topics, like fuzzy algebras, fuzzy graph theory, and fuzzy topologies.

The authors take the reader through the development of the field of fuzzy mathematics, starting with the publication in 1965 of Lotfi Asker Zadeh's seminal paper, Fuzzy Sets.

The book begins with the basics of fuzzy mathematics before moving on to more complex topics, including:

  • Fuzzy sets
  • Fuzzy numbers
  • Fuzzy relations
  • Possibility theory
  • Fuzzy abstract algebra
  • And more

Perfect for advanced undergraduate students, graduate students, and researchers with an interest in the field of fuzzy mathematics, A Modern Introduction to Fuzzy Mathematics walks through both foundational concepts and cutting-edge, new mathematics in the field.

Table of Contents

  1. Cover
  2. Preface
    1. Reference
  3. Acknowledgments
  4. 1 Introduction
    1. 1.1 What Is Vagueness?
    2. 1.2 Vagueness, Ambiguity, Uncertainty, etc.
    3. 1.3 Vagueness and Fuzzy Mathematics
    4. Exercises
    5. Notes
  5. 2 Fuzzy Sets and Their Operations
    1. 2.1 Algebras of Truth Values
    2. 2.2 Zadeh's Fuzzy Sets
    3. 2.3 ‐Cuts of Fuzzy Sets
    4. 2.4 Interval-valued and Type 2 Fuzzy Sets
    5. 2.5 Triangular Norms and Conorms
    6. 2.6 ‐fuzzy Sets
    7. 2.7 “Intuitionistic” Fuzzy Sets and Their Extensions
    8. 2.8 The Extension Principle
    9. 2.9* Boolean‐Valued Sets
    10. 2.10* Axiomatic Fuzzy Set Theory
    11. Exercises
    12. Notes
  6. 3 Fuzzy Numbers and Their Arithmetic
    1. 3.1 Fuzzy Numbers
    2. 3.2 Arithmetic of Fuzzy Numbers
    3. 3.3 Linguistic Variables
    4. 3.4 Fuzzy Equations5
    5. 3.5 Fuzzy Inequalities
    6. 3.6 Constructing Fuzzy Numbers
    7. 3.7 Applications of Fuzzy Numbers
    8. Exercises
    9. Notes
  7. 4 Fuzzy Relations
    1. 4.1 Crisp Relations
    2. 4.2 Fuzzy Relations
    3. 4.3 Cartesian Product, Projections, and Cylindrical Extension
    4. 4.4 New Fuzzy Relations from Old Ones
    5. 4.5 Fuzzy Binary Relations on a Set
    6. 4.6 Fuzzy Orders
    7. 4.7 Elements of Fuzzy Graph Theory
    8. 4.8 Fuzzy Category Theory
    9. 4.9 Fuzzy Vectors5
    10. 4.10 Applications
    11. Exercises
    12. Notes
  8. 5 Possibility Theory
    1. 5.1 Fuzzy Restrictions and Possibility Theory
    2. 5.2 Possibility and Necessity Measures
    3. 5.3 Possibility Theory
    4. 5.4 Possibility Theory and Probability Theory
    5. 5.5 An Unexpected Application of Possibility Theory
    6. Exercises
    7. Note
  9. 6 Fuzzy Statistics
    1. 6.1 Random Variables
    2. 6.2 Fuzzy Random Variables
    3. 6.3 Point Estimation
    4. 6.4 Fuzzy Point Estimation
    5. 6.5 Interval Estimation
    6. 6.6 Interval Estimation for Fuzzy Data
    7. 6.7 Hypothesis Testing
    8. 6.8 Fuzzy Hypothesis Testing
    9. 6.9 Statistical Regression
    10. 6.10 Fuzzy Regression
    11. Exercises
    12. Notes
  10. 7 Fuzzy Logics
    1. 7.1 Mathematical Logic
    2. 7.2 Many‐Valued Logics
    3. 7.3 On Fuzzy Logics
    4. 7.4 Hájek's Basic Many‐Valued Logic
    5. 7.5 Łukasiewicz Fuzzy Logic
    6. 7.6 Product Fuzzy Logic
    7. 7.7 Gödel Fuzzy Logic
    8. 7.8 First‐Order Fuzzy Logics
    9. 7.9 Fuzzy Quantifiers
    10. 7.10 Approximate Reasoning
    11. 7.11 Application: Fuzzy Expert Systems
    12. 7.12 A Logic of Vagueness
    13. Exercises
    14. Notes
  11. 8 Fuzzy Computation
    1. 8.1 Automata, Grammars, and Machines
    2. 8.2 Fuzzy Languages and Grammars
    3. 8.3 Fuzzy Automata
    4. 8.4 Fuzzy Turing Machines
    5. 8.5 Other Fuzzy Models of Computation
  12. 9 Fuzzy Abstract Algebra
    1. 9.1 Groups, Rings, and Fields
    2. 9.2 Fuzzy Groups
    3. 9.3 Abelian Fuzzy Subgroups
    4. 9.4 Fuzzy Rings and Fuzzy Fields
    5. 9.5 Fuzzy Vector Spaces
    6. 9.6 Fuzzy Normed Spaces
    7. 9.7 Fuzzy Lie Algebras
    8. Exercises
    9. Note
  13. 10 Fuzzy Topology
    1. 10.1 Metric and Topological Spaces1
    2. 10.2 Fuzzy Metric Spaces
    3. 10.3 Fuzzy Topological Spaces6
    4. 10.4 Fuzzy Product Spaces
    5. 10.5 Fuzzy Separation
    6. 10.6 Fuzzy Nets
    7. 10.7 Fuzzy Compactness
    8. 10.8 Fuzzy Connectedness
    9. 10.9 Smooth Fuzzy Topological Spaces
    10. 10.10 Fuzzy Banach and Fuzzy Hilbert Spaces
    11. 10.11 Fuzzy Topological Systems
    12. Exercises
    13. Notes
  14. 11 Fuzzy Geometry
    1. 11.1 Fuzzy Points and Fuzzy Distance
    2. 11.2 Fuzzy Lines and Their Properties
    3. 11.3 Fuzzy Circles
    4. 11.4 Regular Fuzzy Polygons
    5. 11.5 Applications in Theoretical Physics
    6. Exercises
  15. 12 Fuzzy Calculus
    1. 12.1 Fuzzy Functions
    2. 12.2 Integrals of Fuzzy Functions
    3. 12.3 Derivatives of Fuzzy Functions
    4. 12.4 Fuzzy Limits of Sequences and Functions
    5. Exercises
  16. A Fuzzy Approximation
    1. A.1 Weierstrass and Stone–Weierstrass Approximation Theorems
    2. A.2 Weierstrass and Stone–Weierstrass Fuzzy Analogs
    3. Note
  17. B Chaos and Vagueness
    1. B.1 Chaos Theory in a Nutshell
    2. B.2 Fuzzy Chaos
    3. B.3 Fuzzy Fractals
    4. Notes
  18. Works Cited
  19. Subject Index
  20. Author Index
  21. End User License Agreement
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