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