Modern Introduction to Fuzzy Mathematics

APOSTOLOS SYROPOULOS, PHD, is an independent scholar based in Xanthi, Greece. His research interests include fuzzy mathematics, the philosophy of vagueness, computability theory, category theory, and digital typography. He has authored or co-authored more than 10 books and more than 60 papers and articles. He has served in the program committee of numerous scientific conferences and has reviewed papers for many journals. THEOPHANES GRAMMENOS, PHD, is Assistant Professor of Applied Mathematics in the Department of Civil Engineering, University of Thessaly, Greece. He received his PhD from University of Athens, Greece. Dr. Grammenos is a member of the editorial board for Applied Mathematics and the International Journal of Applied Mathematical Research.

• Preface VChapter 1 Introduction 11.1 What is vagueness? 11.2 Vagueness, Ambiguity, Uncertainty,… 41.3 Vagueness and Fuzzy Mathematics 5Chapter 2 Fuzzy Sets and Their Operations 92.1 Algebras of Truth Values 92.1.1 Posets 92.1.2 Lattices 112.1.3 Frames 112.2 Zadeh’s Fuzzy Sets 122.3 α-cuts of Fuzzy Sets 162.4 Interval Valued and Type 2 Fuzzy Sets 192.5 Triangular Norms and Conorms 212.6 L-fuzzy Sets 232.7 “Intuitionistic” Fuzzy Sets and Their Extensions 242.8 The Extension Principle 282.9* Boolean-Valued Sets 302.10* Axiomatic Fuzzy Set Theory 32Chapter 3 Fuzzy Numbers and Their Arithmetic 353.1 Fuzzy Numbers 353.1.1 Triangular Fuzzy Numbers 363.1.2 Trapezoidal Fuzzy Numbers 373.1.3 Gaussian Fuzzy Numbers 373.1.4 Quadratic Fuzzy Numbers 393.1.5 Exponential Fuzzy Numbers 413.1.6 LR Fuzzy Numbers 413.1.7 Generalized Fuzzy Numbers 423.2 Arithmetic of Fuzzy Numbers 433.2.1 Interval Arithmetic 433.2.2 Interval Arithmetic and α-Cuts 433.2.3 Fuzzy Arithmetic and the Extension Principle 443.2.4 Fuzzy Arithmetic of Triangular Fuzzy Numbers 453.2.5 Fuzzy Arithmetic of Generalized Fuzzy Numbers 453.2.6 Comparing Fuzzy Numbers 473.3 Linguistic Variables 493.4 Fuzzy Equations 503.4.1 Solving the Fuzzy Equation 𝐴 ⋅ 𝑋 + 𝐵 = 𝐶 503.4.2 Solving the Fuzzy Equation 𝐴 ⋅ 𝑋2 + 𝐵 ⋅ 𝑋 + 𝐶 = 𝐷 533.5 Fuzzy Inequalities 553.6 Constructing Fuzzy Numbers 553.7 Applications of Fuzzy Numbers 573.7.1 Simulation of the Human Glucose Metabolism 573.7.2 Estimation of an Ongoing Project’s Completion Time 60Chapter 4 Fuzzy Relations 654.1 Crisp Relations 654.2 Fuzzy Relations 694.3 Cartesian Product, Projections, and Cylindrical Extension 704.4 New Fuzzy Relations From Old Ones 724.5 Fuzzy Binary Relations on a Set 754.6 Fuzzy Orders 804.7 Elements of Fuzzy Graph Theory 824.7.1 Graphs and Hypergraphs 824.7.2 Fuzzy Graphs 834.7.3 Fuzzy Hypergraphs 874.8* Fuzzy Category Theory 894.9* Fuzzy Vectors 964.10 Applications 97Chapter 5 Possibility Theory 1015.1 Fuzzy Restrictions and Possibility Theory 1015.2 Possibility and Necessity Measures 1035.3 Possibility Theory 1055.4 Possibility Theory and Probability Theory 1085.5 An Unexpected Application of Possibility Theory 110Chapter 6 Fuzzy Statistics 1176.1 Random Variables 1176.2 Fuzzy Random Variables 1206.3 Point Estimation 1236.3.1 The unbiased estimator 1246.3.2 The consistent estimator 1256.3.3 The maximum likelihood estimator 1266.4 Fuzzy Point Estimation 1276.5 Interval Estimation 1286.6 Interval Estimation for Fuzzy Data 1296.7 Hypothesis Testing 1316.8 Fuzzy Hypothesis Testing 1326.9 Statistical Regression 1346.10 Fuzzy Regression 136Chapter 7 Fuzzy Logics 1417.1 Mathematical Logic 1417.2 Many-Valued Logics 1467.3 On Fuzzy Logics 1517.4 Hájek’s Basic Many-Valued Logic 1527.5 Łukasiewicz Fuzzy Logic 1557.6 Product Fuzzy Logic 1577.7 Gödel Fuzzy Logic 1587.8 First Order Fuzzy Logics 1607.9 Fuzzy Quantifiers 1627.10 Approximate Reasoning 1637.11 Application: Fuzzy Expert Systems 1667.12* A Logic of Vagueness 171Chapter 8 Fuzzy Computation 1738.1 Automata, Grammars, and Machines 1738.2 Fuzzy Languages and Grammars 1788.3 Fuzzy Automata 1818.4 Fuzzy Turing Machines 1868.5 Other Fuzzy Models of Computation 190Chapter 9 Fuzzy Abstract Algebra 1959.1 Groups, Rings, and Fields 1959.2 Fuzzy Groups 1999.3 Abelian Fuzzy Subgroups 2049.4 Fuzzy Rings and Fuzzy Fields 2069.5 Fuzzy Vector Spaces 2089.6 Fuzzy Normed Spaces 2099.7 Fuzzy Lie Algebras 210Chapter 10 Fuzzy Topology 21310.1 Metric and Topological Spaces 21310.2 Fuzzy Metric Spaces 21810.3 Fuzzy Topological Spaces 22110.4 Fuzzy Product Spaces 22410.5 Fuzzy Separation 22610.5.1 Separation 23110.6 Fuzzy Nets 23110.7 Fuzzy Compactness 23210.8 Fuzzy Connectedness 23310.9 Smooth Fuzzy Topological Spaces 23410.10 Fuzzy Banach and Fuzzy Hilbert Spaces 23510.11* Fuzzy Topological Systems 238Chapter 11 Fuzzy Geometry 24311.1 Fuzzy Points and Fuzzy Distance 24311.2 Fuzzy Lines and their Properties 24611.3 Fuzzy Circles 24911.4 Regular Fuzzy Polygons 25211.5 Applications in Theoretical Physics 256Chapter 12 Fuzzy Calculus 25912.1 Fuzzy Functions 25912.2 Integrals of Fuzzy Functions 26312.3 Derivatives of Fuzzy Functions 26612.4 Fuzzy Limits of Sequences and Functions 26912.4.1 Fuzzy Ordinary Differential Equations 27212.4.2 Fuzzy Partial Differential Equations 277Appendix A Fuzzy Approximation 283A.1 Weierstrass and Stone-Weierstrass Approximation Theorems 283A.2 Weierstrass and Stone-Weierstrass Fuzzy Analogues 284Appendix B Chaos and Vagueness 287B.1 Chaos Theory in a Nutshell 287B.2 Fuzzy Chaos 289B.3 Fuzzy Fractals 291Works Cited 293Subject Index 311Name Index 323