First Course on Fuzzy Theory and Applications

Kwang H. Lee
Table of Contents
Chapter 1.
FUZZY SETS ------------------------------------------------------------- 1
1.1 Sets ------------------------------------------------------------------------- 1
1.2 Operation of Sets ------------------------------------------------------------- 3
1.3 Characteristics of Crisp Set ------------------------------------------------- 5
1.4 Definition of Fuzzy Set ------------------------------------------------------ 7
1.5 Expanding Concepts of Fuzzy Set ---------------------------------------- 14
1.6 Standard Operation of Fuzzy Set ----------------------------------------- 21
[SUMMARY] -------------------------------------------------------------------- 22
[EXERCISES] ------------------------------------------------------------------- 24
Chapter 2.
THE OPERATION OF FUZZY SET --------------------------------- 27
2.1 Standard Operations of Fuzzy Set ---------------------------------------- 27
2.2 Fuzzy Complement --------------------------------------------------------- 28
2.3 Fuzzy Union ----------------------------------------------------------------- 32
2.4 Fuzzy Intersection ---------------------------------------------------------- 35
2.5 Other Operations In Fuzzy Set -------------------------------------------- 38
2.6 T-norms and T-conorms --------------------------------------------------- 45
[SUMMARY] ------------------------------------------------------------------ 47
[EXERCISES] ------------------------------------------------------------------- 51
Chapter 3.
FUZZY RELATION AND COMPOSITION ------------------------ 53
3.1 Crisp Relation --------------------------------------------------------------- 53
3.2 Properties of Relation on a Single Set ---------------------------------- 62
3.3 Fuzzy Relation -------------------------------------------------------------- 68
3.4 Extension of Fuzzy Set ----------------------------------------------------- 80
[SUMMARY] -------------------------------------------------------------------- 86
[EXERCISES] ------------------------------------------------------------------- 88
Chapter 4.
FUZZY GRAPH AND RELATION ---------------------------------- 91
4.1 Fuzzy Graph ----------------------------------------------------------------- 91
4.2 Characteristics of Fuzzy Relation --------------------------------------- 103
4.3 Classification of Fuzzy Relation ---------------------------------------- 108
4.4 Other Fuzzy Relations ---------------------------------------------------- 116
[SUMMARY] ------------------------------------------------------------------ 124
[EXERCISES] ------------------------------------------------------------------ 126
Chapter 5.
FUZZY NUMBER ----------------------------------------------------- 129
5.1 Concept of Fuzzy Number ----------------------------------------------- 129
5.2 Operation of Fuzzy Number --------------------------------------------- 132
5.3 Triangular Fuzzy Number ------------------------------------------------ 137
5.4 Other Types of Fuzzy Number ------------------------------------------ 145
[SUMMARY] ------------------------------------------------------------------ 149
[EXERCISES] ------------------------------------------------------------------ 150
Chapter 6.
FUZZY FUNCTION -------------------------------------------------- 153
6.1 Kinds of Fuzzy Function ------------------------------------------------- 153
6.2 Fuzzy Extrema of Function ---------------------------------------------- 158
6.3 Integration and Differenciation of Fuzzy Function ------------------- 163
[SUMMARY] ------------------------------------------------------------------ 168
[EXERCISES] ------------------------------------------------------------------ 169
Chapter 7.
PROBABILITY AND UNCERTAINTY --------------------------- 171
7.1 Probability and Possibility ----------------------------------------------- 171
7.2 Fuzzy Event ---------------------------------------------------------------- 174
7.3 Uncertainty ----------------------------------------------------------------- 179
7.4 Measure of Fuzziness ----------------------------------------------------- 181
[SUMMARY] ------------------------------------------------------------------ 189
[EXERCISES] ------------------------------------------------------------------ 190
Chapter 8.
FUZZY LOGIC -------------------------------------------------------- 193
8.1 Classical Logic ------------------------------------------------------------ 193
8.2 Fuzzy Logic ---------------------------------------------------------------- 201
8.3 Linguistic Variable -------------------------------------------------------- 204
8.4 Fuzzy Truth Qualifier ---------------------------------------------------- 206
8.5 Representation of Fuzzy Rule ------------------------------------------- 210
[SUMMARY] ------------------------------------------------------------------ 213
[EXERCISES] ------------------------------------------------------------------ 215
Chapter 9.
FUZZY INFERENCE ------------------------------------------------- 217
9.1 Composition of Rules ----------------------------------------------------- 217
9.2 Fuzzy Rules and Implication -------------------------------------------- 221
9.3 Inference Mechanism ----------------------------------------------------- 224
9.4 Inference Methods -------------------------------------------------------- 236
[SUMMARY] ------------------------------------------------------------------ 247
[EXERCISES] ------------------------------------------------------------------ 250
Chapter 10.
FUZZY CONTROL AND FUZZY EXPERT SYSTEMS ------- 253
10.1 Fuzzy Logic Controller ------------------------------------------------ 253
10.2 Fuzzification Interface Component ----------------------------------- 255
10.3 Knowledge Base Component ------------------------------------------ 257
10.4 Inference (Decision Making Logic) ----------------------------------- 265
10.5 Defuzzification ----------------------------------------------------------- 269
10.6 Design Procedure of Fuzzy Logic Controller ------------------------ 272
10.7 Application Example of FLC Design --------------------------------- 273
10.8 Fuzzy Expert Systems --------------------------------------------------- 277
[SUMMARY] ------------------------------------------------------------------ 280
[EXERCISES] ------------------------------------------------------------------ 282
Chapter 11.
FUSION OF FUZZY SYSTEM AND NEURAL NETWORKS ------- 285
11.1 Neural Networks --------------------------------------------------------- 285
11.2 Fusion With Neural Networks ----------------------------------------- 290
[SUMMARY] ------------------------------------------------------------------ 306
[EXERCISES] ------------------------------------------------------------------ 308
Chapter 12.
FUSION OF FUZZY SYSTEM AND GENETIC ALGORITHMS ---- 309
12.1 Genetic Algorithms ------------------------------------------------------ 309
12.2 Fusion With Genetic Algorithms -------------------------------------- 314
[SUMMARY] ------------------------------------------------------------------ 323
[EXERCISES] ------------------------------------------------------------------ 324
BIBLIOGRAPH -------------------------------------------------------- 325
INDEX ------------------------------------------------------------------- 333

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