## Table of Contents

**Foreword.**

**Preface.**

**I. THEORY.**

**1. From Classical (Crisp) Sets to Fuzzy Sets: A Grand Paradigm Shift.**

Introduction. Crisp Sets: An Overview. Fuzzy Sets: Basic Types. Fuzzy Sets: Basic Concepts. Characteristics and Significance of the Paradigm Shift. Notes. Exercises.

**2. Fuzzy Sets versus Crisp Sets.**

Additional Properties of -Cuts. Representations of Fuzzy Sets. Extension Principle for Fuzzy Sets. Notes. Exercises.

**3. Operations on Fuzzy Sets.**

Types of Operations. Fuzzy Complements. Fuzzy Intersections: t- Norms. Fuzzy Unions: t-Conorms. Combinations of Operations. Aggregation Operations. Notes. Exercises.

**4. Fuzzy Arithmetic.**

Fuzzy Numbers. Linguistic Variables. Arithmetic Operations on Intervals. Arithmetic Operations on Fuzzy Numbers. Lattice of Fuzzy Numbers. Fuzzy Equations. Notes. Exercises.

**5. Fuzzy Relations.**

Crisp versus Fuzzy Relations. Projections and Cylindric Extensions. Binary Fuzzy Relations. Binary Relations on a Single Set. Fuzzy Equivalence Relations. Fuzzy Compatibility Relations. Fuzzy Ordering Relations. Fuzzy Morphisms. Sup-i Compositions of Fuzzy Relations. Inf- Compositions of Fuzzy Relations. Notes. Exercises.

**6. Fuzzy Relation Equations.**

General Discussion. Problem Partitioning. Solution Method. Fuzzy Relation Equations Based on Sup-i Compositions. Fuzzy Relation Equations Based on Inf-Compositions. Approximate Solutions. The Use of Neural Networks. Notes. Exercises.

**7. Possibility Theory.**

Fuzzy Measures. Evidence Theory. Possibility Theory. Fuzzy Sets and Possibility Theory. Possibility Theory versus Probability Theory. Notes. Exercises.

**8. Fuzzy Logic.**

Classical Logic: An Overview. Multivalued Logics. Fuzzy Propositions. Fuzzy Quantifiers. Linguistic Hedges. Inference from Conditional Fuzzy Propositions. Inference from Conditional and Qualified Propositions. Inference from Quantified Propositions. Notes. Exercises.

**9. Uncertainty-Based Information.**

Information and Uncertainty. Nonspecificity of Crisp Sets. Nonspecificity of Fuzzy Sets. Fuzziness of Fuzzy Sets. Uncertainty in Evidence Theory. Summary of Uncertainty Measures. Principles of Uncertainty. Notes. Exercises.

**II. APPLICATIONS.**

**10. Constructing Fuzzy Sets and Operations on Fuzzy Sets.**

General Discussion. Methods of Construction: An Overview. Direct Methods with One Expert. Direct Methods with Multiple Experts. Indirect Methods with One Expert. Indirect Methods with Multiple Experts. Constructions from Sample Data. Notes. Exercises.

**11. Approximate Reasoning.**

Fuzzy Expert Systems: An Overview. Fuzzy Implications. Selection of Fuzzy Implications. Multiconditional Approximate Reasoning. The Role of Fuzzy Relation Equations. Interval-Valued Approximate Reasoning. Notes. Exercises.

**12. Fuzzy Systems.**

General Discussion. Fuzzy Controllers: An Overview. Fuzzy Controllers: An Example. Fuzzy Systems and Neural Networks. Fuzzy Neural Networks. Fuzzy Automata. Fuzzy Dynamic Systems. Notes. Exercises.

**13. Pattern Recognition.**

Introduction. Fuzzy Clustering. Fuzzy Pattern Recognition. Fuzzy Image Processing. Notes. Exercises.

**14. Fuzzy Databases and Information Retrieval Systems.**

General Discussion. Fuzzy Databases. Fuzzy Information Retrieval. Notes. Exercises.

**15. Fuzzy Decision Making.**

General Discussion. Individual Decision Making. Multiperson Decision Making. Multicriteria Decision Making. Multistage Decision Making. Fuzzy Ranking Methods. Fuzzy Linear Programming. Notes. Exercises.

**16. Engineering Applications.**

Introduction. Civil Engineering. Mechanical Engineering. Industrial Engineering. Computer Engineering. Reliability Theory. Robotics. Notes. Exercises.

**17. Miscellaneous Applications.**

Introduction. Medicine. Economics. Fuzzy Systems and Genetic Algorithms. Fuzzy Regression. Interpersonal Communication. Other Applications. Notes. Exercises.

**Appendix A. Neural Networks: An Overview.**

**Appendix B. Genetic Algorithms: An Overview.**

**Appendix C. Rough Sets versus Fuzzy Sets.**

**Appendix D. Proofs of Some Mathematical Theorems.**

**Appendix E. Glossary of Key Concepts.**

**Appendix F. Glossary of Symbols.**

**Bibliography.**

**Bibliographical Index.**

**Name Index.**

**Subject Index.**

Fuzzy Sets and Fuzzy Logic: Theory and Applications

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