## First Course on Fuzzy Theory and ApplicationsFuzzy theory has become a subject that generates much interest among the courses for graduate students. However, it was not easy to find a suitable textbook to use in the introductory course and to recommend to the students who want to self-study. The main purpose of this book is just to meet that need. The author has given lectures on the fuzzy theory and its applications for ten years and continuously developed lecture notes on the subject. This book is a publication of the modification and summary of the lecture notes. The fundamental idea of the book is to provide basic and concrete concepts of the fuzzy theory and its applications, and thus the author focused on easy illustrations of the basic concepts. There are numerous examples and figures to help readers to understand and also added exercises at the end of each chapter. This book consists of two parts: a theory part and an application part. The first part (theory part) includes chapters from 1 to 8. Chapters 1 and 2 introduce basic concepts of fuzzy sets and operations, and Chapters 3 and 4 deal with the multi-dimensional fuzzy sets. Chapters 5 and 6 are extensions of the fuzzy theory to the number and function, and Chapters 7 and 8 are developments of fuzzy properties on the probability and logic theories. |

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### Contents

Chapter 1 FUZZY SETS | 1 |

12 Operation of Sets | 3 |

13 Characteristics of Crisp Set | 5 |

14 Definition of Fuzzy Set | 7 |

15 Expanding Concepts of Fuzzy Set | 14 |

16 Standard Operation of Fuzzy Set | 21 |

SUMMARY | 22 |

EXERCISES | 24 |

72 Fuzzy Event | 174 |

73 Uncertainty | 179 |

74 Measure of Fuzziness | 181 |

SUMMARY | 189 |

EXERCISES | 190 |

Chapter 8 FUZZY LOGIC | 193 |

82 Fuzzy Logic | 201 |

83 Linguistic Variable | 204 |

Chapter 2 THE OPERATION OF FUZZY SET | 27 |

22 Fuzzy Complement | 28 |

23 Fuzzy Union | 32 |

24 Fuzzy Intersection | 35 |

25 Other Operations in Fuzzy Set | 38 |

26 tnorms and tconorms | 45 |

SUMMARY | 47 |

EXERCISES | 51 |

Chapter 3 FUZZY RELATION AND COMPOSITION | 53 |

32 Properties of Relation on A Single Set | 62 |

33 Fuzzy Relation | 68 |

34 Extension of Fuzzy Set | 80 |

SUMMARY | 86 |

EXERCISES | 88 |

Chapter 4 FUZZY GRAPH AND RELATION | 91 |

42 Characteristics of Fuzzy Relation | 103 |

43 Classification of Fuzzy Relation | 108 |

44 Other Fuzzy Relations | 116 |

SUMMARY | 124 |

EXERCISES | 126 |

Chapter 5 FUZZY NUMBER | 129 |

52 Operation of Fuzzy Number | 132 |

53 Triangular Fuzzy Number | 137 |

54 Other Types of Fuzzy Number | 145 |

SUMMARY | 149 |

EXERCISES | 150 |

Chapter 6 FUZZY FUNCTION | 153 |

62 Fuzzy Extrema of Function | 158 |

63 Integration and Differenciation of Fuzzy Function | 163 |

SUMMARY | 168 |

EXERCISES | 169 |

Chapter 7 PROBABILISY AND UNCERTAINTY | 171 |

84 Fuzzy Truth Qualifier | 206 |

85 Representation of Fuzzy Rule | 210 |

SUMMARY | 213 |

EXERCISES | 215 |

Chapter 9 FUZZY INFERENCE | 217 |

92 Fuzzy Rules and Implication | 221 |

93 Inference Mechanism | 224 |

94 Inference Methods | 236 |

SUMMARY | 247 |

EXERCISES | 250 |

Chapter 10 FUZZY CONTROL AND FUZZY EXPERT SYSTEMS | 253 |

102 Fuzzification Interface Component | 255 |

103 Knowledge Base Component | 257 |

104 Inference Decision Making Logic | 265 |

105 Defuzzification | 269 |

106 Design Procedure of Fuzzy Logic Controller | 272 |

107 Application Example of FLC Design | 273 |

108 Fuzzy Expert Systems | 277 |

SUMMARY | 280 |

EXERCISES | 282 |

Chapter 11 FUSION OF FUZZY SYSTEM AND NEURAL NETWORKS | 285 |

112 Fusion with Neural Networks | 290 |

SUMMARY | 306 |

EXERCISE | 308 |

Chapter 12 FUSION OF FUZZY SYSTEMS AND GENETIC ALGORITHMS | 309 |

122 Fusion with Genetic Algorithms | 314 |

SUMMARY | 323 |

EXERCISE | 324 |

325 | |

333 | |

### Common terms and phrases

a-cut set Axiom Cartesian product chapter chromosomes compatibility relation complement set Composition of fuzzy control action crisp function crisp relation crisp set crossover defined as follows Definition Fuzzy defuzzification denoted Determine disjunctive sum domain elements equivalence relation evaluation fuzzifying function fuzzy control rules fuzzy event fuzzy graph fuzzy implication fuzzy inference fuzzy logic fuzzy logic controller fuzzy order relation fuzzy partition fuzzy relation fuzzy rules fuzzy set fuzzy systems genetic algorithms given Hamming distance inference method input variables intersection layer Lemma linguistic terms linguistic variable logic formula lookup table matching degree max-min maximum value membership degree membership function neural networks node obtained Operation of Fuzzy parameters possibility probability distribution proposition real number Reflexive relation represented result rule base shown in Fig singleton input subset Symmetric relation t-conorm t-norm Transitive relation trapezoidal fuzzy number triangular fuzzy number truth value universal set universe of discourse