Legacy Documentation

Contents

0.1 Introduction

0.1.1 Fuzzy Logic 0.1.2 The Manual 0.1.3 The Demonstration Notebooks 0.1.4 The Packages 0.1.5 Loading the Package 0.1.6 Getting More Information about Mathematica 0.1.7 Getting More Information about Fuzzy Sets 0.1.8 Acknowledgments 0.1.9 About the Authors

0.2 New Features in This Release

0.2.1 Introduction 0.2.2 New Membership Functions 0.2.3 Fuzzy Graph 0.2.4 Defuzzifications 0.2.5 Additional Operators 0.2.6 New Intersections and Unions 0.2.7 Alpha Cuts for Fuzzy Relations 0.2.8 Fuzzy Relation Equations 0.2.9 Random Fuzzy Sets and Fuzzy Relations 0.2.10 Fuzzy Inferencing 0.2.11 Fuzzy Arithmetic 0.2.12 Fuzzy Clustering

Part 1 Manual

1.1 Creating Fuzzy Sets

1.1.1 Introduction 1.1.2 Basic Objects 1.1.3 Functions for Creating Fuzzy Sets

1.2 Creating Fuzzy Relations

1.2.1 Introduction 1.2.2 Basic Objects 1.2.3 Functions for Creating Fuzzy Relations

1.3 Fuzzy Operations

1.3.1 Introduction 1.3.2 Fuzzy Set Operations 1.3.3 Fuzzy Relation Operations 1.3.4 Fuzzy Set or Fuzzy Relation Operations

1.4 Aggregation Operations

1.4.1 Introduction 1.4.2 Intersection (t-norm) and Union (s-norm) Operations 1.4.3 Averaging Operations 1.4.4 Difference Operations 1.4.5 User-Defined Aggregators

1.5 Fuzzy Set Visualization

1.5.1 Introduction 1.5.2 Visualization Functions

1.6 Fuzzy Relation Visualization

1.6.1 Introduction 1.6.2 Visualization Functions

1.7 Compositions

1.7.1 Introduction 1.7.2 Composition Function

1.8 Fuzzy Inferencing

1.8.1 Introduction 1.8.2 Inference Functions 1.8.3 Composition-Based Inference 1.8.4 Rule-Based Inference

1.9 Fuzzy Arithmetic

1.9.1 Introduction 1.9.2 Fuzzy Arithmetic Functions

1.10 Discrete Fuzzy Arithmetic

1.10.1 Introduction 1.10.2 Discrete Arithmetic on Triangular Fuzzy Numbers 1.10.3 Discrete Arithmetic on Gaussian Fuzzy Numbers

1.11 ukasiewicz Sets and Logic

1.11.1 Introduction 1.11.2 Creating ukasiewicz Sets 1.11.3 Operations on Ln Sets

1.12 Fuzzy Clustering

1.12.1 Introduction 1.12.2 Fuzzy C-Means Clustering (FCM) 1.12.3 Example

Appendix

Fuzzy Operator Formulas Intersection Formulas Union Formulas Averaging Formulas Miscellaneous Formulas

Bibliography

Part 2 Demonstration Notebooks

2.1 Sets Versus Fuzzy Sets

2.1.1 Introduction 2.1.2 Characteristic Function and Membership Function 2.1.3 Graphic Interpretation of Sets References

2.2 Standard Operations

2.2.1 Introduction 2.2.2 Fuzzy Operations References

2.3 Fuzzy Relations

2.3.1 Introduction 2.3.2 Fuzzy Relation Form 2.3.3 Projection of a Fuzzy Relation 2.3.4 Fuzzy Operations 2.3.5 Composition of Two Fuzzy Relations 2.3.6 Binary Relations References

2.4 Fuzzy Modeling

2.4.1 Introduction 2.4.2 Representing the Model Input 2.4.3 Representing the Model Output 2.4.4 Creating Linguistic Control Rules 2.4.5 Building the Model 2.4.6 Using the Model 2.4.7 Evaluating the Model References

2.5 Fuzzy Logic Control

2.5.1 Introduction 2.5.2 Defining Input Membership Functions 2.5.3 Defining Output Membership Functions 2.5.4 Defining Control Rules 2.5.5 Simulation Functions 2.5.6 Test Run 1 2.5.7 Test Run 2 2.5.8 Control Surface References

2.6 Fuzzy Numbers

2.6.1 Introduction 2.6.2 Creating Fuzzy Numbers 2.6.3 Fuzzy Arithmetic References

2.7 Digital Fuzzy Sets and Multivalued Logic

2.7.1 Introduction 2.7.2 Creating Digital Fuzzy Sets 2.7.3 ukasiewicz Multivalued Logic References

2.8 Additional Examples

2.8.1 Introduction 2.8.2 Example 1: Classifying Houses 2.8.3 Example 2: Representing Age 2.8.4 Example 3: Finding the Disjunctive Sum 2.8.5 Example 4: Natural Numbers 2.8.6 Example 5: Fuzzy Hedges 2.8.7 Example 6: Distance Relation 2.8.8 Example 7: Choosing a Job 2.8.9 Example 8: Digital Fuzzy Sets 2.8.10 Example 9: Image Processing References

Index