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Contents

0.1 Introduction

0.1.1 Fuzzy Logic Bullet0.1.2 The Manual Bullet0.1.3 The Demonstration Notebooks Bullet0.1.4 The Packages Bullet0.1.5 Loading the Package Bullet0.1.6 Getting More Information about Mathematica Bullet0.1.7 Getting More Information about Fuzzy Sets Bullet0.1.8 Acknowledgments Bullet0.1.9 About the Authors

0.2 New Features in This Release

0.2.1 Introduction Bullet0.2.2 New Membership Functions Bullet0.2.3 Fuzzy Graph Bullet0.2.4 Defuzzifications Bullet0.2.5 Additional Operators Bullet0.2.6 New Intersections and Unions Bullet0.2.7 Alpha Cuts for Fuzzy Relations Bullet0.2.8 Fuzzy Relation Equations Bullet0.2.9 Random Fuzzy Sets and Fuzzy Relations Bullet0.2.10 Fuzzy Inferencing Bullet0.2.11 Fuzzy Arithmetic Bullet0.2.12 Fuzzy Clustering

Part 1 Manual

1.1 Creating Fuzzy Sets

1.1.1 Introduction Bullet1.1.2 Basic Objects Bullet1.1.3 Functions for Creating Fuzzy Sets

1.2 Creating Fuzzy Relations

1.2.1 Introduction Bullet1.2.2 Basic Objects Bullet1.2.3 Functions for Creating Fuzzy Relations

1.3 Fuzzy Operations

1.3.1 Introduction Bullet1.3.2 Fuzzy Set Operations Bullet1.3.3 Fuzzy Relation Operations Bullet1.3.4 Fuzzy Set or Fuzzy Relation Operations

1.4 Aggregation Operations

1.4.1 Introduction Bullet1.4.2 Intersection (t-norm) and Union (s-norm) Operations Bullet1.4.3 Averaging Operations Bullet1.4.4 Difference Operations Bullet1.4.5 User-Defined Aggregators

1.5 Fuzzy Set Visualization

1.5.1 Introduction Bullet1.5.2 Visualization Functions

1.6 Fuzzy Relation Visualization

1.6.1 Introduction Bullet1.6.2 Visualization Functions

1.7 Compositions

1.7.1 Introduction Bullet1.7.2 Composition Function

1.8 Fuzzy Inferencing

1.8.1 Introduction Bullet1.8.2 Inference Functions Bullet1.8.3 Composition-Based Inference Bullet1.8.4 Rule-Based Inference

1.9 Fuzzy Arithmetic

1.9.1 Introduction Bullet1.9.2 Fuzzy Arithmetic Functions

1.10 Discrete Fuzzy Arithmetic

1.10.1 Introduction Bullet1.10.2 Discrete Arithmetic on Triangular Fuzzy Numbers Bullet1.10.3 Discrete Arithmetic on Gaussian Fuzzy Numbers

1.11 CapitalLSlashukasiewicz Sets and Logic

1.11.1 Introduction Bullet1.11.2 Creating CapitalLSlashukasiewicz Sets Bullet1.11.3 Operations on Ln Sets

1.12 Fuzzy Clustering

1.12.1 Introduction Bullet1.12.2 Fuzzy C-Means Clustering (FCM) Bullet1.12.3 Example

Appendix

Fuzzy Operator Formulas BulletIntersection Formulas BulletUnion Formulas BulletAveraging Formulas BulletMiscellaneous Formulas

Bibliography

Part 2 Demonstration Notebooks

2.1 Sets Versus Fuzzy Sets

2.1.1 Introduction Bullet2.1.2 Characteristic Function and Membership Function Bullet2.1.3 Graphic Interpretation of Sets BulletReferences

2.2 Standard Operations

2.2.1 Introduction Bullet2.2.2 Fuzzy Operations BulletReferences

2.3 Fuzzy Relations

2.3.1 Introduction Bullet2.3.2 Fuzzy Relation Form Bullet2.3.3 Projection of a Fuzzy Relation Bullet2.3.4 Fuzzy Operations Bullet2.3.5 Composition of Two Fuzzy Relations Bullet2.3.6 Binary Relations BulletReferences

2.4 Fuzzy Modeling

2.4.1 Introduction Bullet2.4.2 Representing the Model Input Bullet2.4.3 Representing the Model Output Bullet2.4.4 Creating Linguistic Control Rules Bullet2.4.5 Building the Model Bullet2.4.6 Using the Model Bullet2.4.7 Evaluating the Model BulletReferences

2.5 Fuzzy Logic Control

2.5.1 Introduction Bullet2.5.2 Defining Input Membership Functions Bullet2.5.3 Defining Output Membership Functions Bullet2.5.4 Defining Control Rules Bullet2.5.5 Simulation Functions Bullet2.5.6 Test Run 1 Bullet2.5.7 Test Run 2 Bullet2.5.8 Control Surface BulletReferences

2.6 Fuzzy Numbers

2.6.1 Introduction Bullet2.6.2 Creating Fuzzy Numbers Bullet2.6.3 Fuzzy Arithmetic BulletReferences

2.7 Digital Fuzzy Sets and Multivalued Logic

2.7.1 Introduction Bullet2.7.2 Creating Digital Fuzzy Sets Bullet2.7.3 CapitalLSlashukasiewicz Multivalued Logic BulletReferences

2.8 Additional Examples

2.8.1 Introduction Bullet2.8.2 Example 1: Classifying Houses Bullet2.8.3 Example 2: Representing Age Bullet2.8.4 Example 3: Finding the Disjunctive Sum Bullet2.8.5 Example 4: Natural Numbers Bullet2.8.6 Example 5: Fuzzy Hedges Bullet2.8.7 Example 6: Distance Relation Bullet2.8.8 Example 7: Choosing a Job Bullet2.8.9 Example 8: Digital Fuzzy Sets Bullet2.8.10 Example 9: Image Processing BulletReferences

Index

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