First Course on Fuzzy Theory and Applications (Advances in Intelligent and Soft Computing 27)

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This basic textbook gives an easily accessible introduction to fuzzy theory and its applications. It provides basic and concrete concepts of the field in a self-contained, condensed and understandable style. This First Course on Fuzzy Theory and Applications includes numerous examples, descriptive illustrations and figures of the basic concepts, as well as exercises at the end of each chapter. The author has long time experience in teaching on fuzzy theory and its applications and continuously developed and summarized his didactic lecture notes into this book. This book can be used in introductory graduate and undergraduate courses in fuzziness and soft computing and is recommendable to students, scientists, engineers, or professionals in the field for self-study.

Author(s): Kwang Hyung Lee
Series: Advances in Intelligent and Soft Computing 27
Edition: 1
Publisher: Springer
Year: 2004

Language: English
Pages: 348
Tags: Математика;Математическая логика;Нечеткая логика;

FIRST COURSE ON FUZZY THEORY AND APPLICATIONS......Page 1
Springerlink......Page 0
Half-title......Page 2
Advances in Soft Computing......Page 3
Title Page......Page 4
Copyright Page......Page 5
Preface......Page 7
Table of Contents......Page 8
1.1.1 Elements of Sets......Page 12
1.1.3 Membership......Page 13
1.2.2 Union......Page 14
1.2.4 Partition of Set......Page 15
1.3.1 Ordinary Characteristics......Page 16
1.3.2 Convex Set......Page 17
1.4.1 Expression for Fuzzy Set......Page 18
1.4.2 Examples of Fuzzy Set......Page 21
1.4.3 Expansion of Fuzzy Set......Page 23
1.5.1 Example of Fuzzy Set......Page 25
1.5.2 α-Cut Set......Page 26
1.5.3 Convex Fuzzy Set......Page 28
1.5.5 The Magnitude of Fuzzy Set......Page 29
1.5.6 Subset of fuzzy set......Page 31
1.6.3 Intersection......Page 32
Summary......Page 33
Exercises......Page 35
2.1 Standard Operations of Fuzzy Set......Page 38
2.2.1 Requirements for Complement Function......Page 39
2.2.2 Example of Complement Function......Page 40
2.3.1 Axioms for Union Function......Page 43
2.3.2 Examples of Union Function......Page 44
2.3.3 Other Union Operations......Page 45
2.4.1 Axioms for Intersection Function......Page 46
2.4.2 Examples of Intersection......Page 47
2.5.1 Disjunctive Sum......Page 49
2.5.2 Difference in Fuzzy Set......Page 52
2.5.3 Distance in Fuzzy Set......Page 53
2.5.4 Cartesian Product of Fuzzy Set......Page 55
2.6.1 Definitions for t-norms and t-conorms......Page 56
2.6.2 Duality of t-norms and t-conorms......Page 57
Summary......Page 58
Exercises......Page 62
3.1.1 Product Set......Page 64
3.1.2 Definition of Relation......Page 65
3.1.3 Characteristics of Relation......Page 67
3.1.4 Representation Methods of Relations......Page 70
3.1.5 Operations on Relations......Page 71
3.1.6 Path and Connectivity in Graph......Page 72
3.2.1 Fundamental Properties......Page 73
3.2.2 Equivalence Relation......Page 74
3.2.3 Compatibility Relation (Tolerance Relation)......Page 75
3.2.4 Pre-order Relation......Page 76
3.2.5 Order Relation......Page 77
3.3.1 Definition of Fuzzy Relation......Page 79
3.3.2 Examples of Fuzzy Relation......Page 81
3.3.3 Fuzzy Matrix......Page 82
3.3.4 Operation of Fuzzy Relation......Page 84
3.3.5 Composition of Fuzzy Relation......Page 86
3.3.6 α-cut of Fuzzy Relation......Page 87
3.3.7 Projection and Cylindrical Extension......Page 88
3.4.1 Extension by Relation......Page 91
3.4.2 Extension Principle......Page 92
3.4.3 Extension by Fuzzy Relation......Page 93
3.4.4 Fuzzy Distance between Fuzzy Sets......Page 94
Summary......Page 97
Exercises......Page 99
4.1.1 Graph and Fuzzy Graph......Page 102
4.1.2 Fuzzy Graph and Fuzzy Relation......Page 103
4.1.3 α-cut of Fuzzy Graph......Page 107
4.1.4 Fuzzy Network......Page 113
4.2.1 Reflexive Relation......Page 114
4.2.2 Symmetric Relation......Page 115
4.2.3 Transitive Relation......Page 116
4.2.4 Transitive Closure......Page 117
4.3.1 Fuzzy Equivalence Relation......Page 119
4.3.2 Fuzzy Compatibility Relation......Page 122
4.3.3 Fuzzy Pre-order Relation......Page 123
4.3.4 Fuzzy Order Relation......Page 124
4.4.1 Fuzzy Ordinal Relation......Page 127
4.4.2 Dissimilitude Relation......Page 129
4.4.3 Fuzzy Morphism......Page 131
4.4.4 Examples of Fuzzy Morphism......Page 132
Summary......Page 135
Exercises......Page 137
5.1.1 Interval......Page 140
5.1.2 Fuzzy Number......Page 141
5.1.3 Operation of Interval......Page 142
5.2.1 Operation of α-cut Interval......Page 143
5.2.2 Operation of Fuzzy Number......Page 144
5.2.3 Examples of Fuzzy Number Operation......Page 145
5.3.1 Definition of Triangular Fuzzy Number......Page 148
5.3.2 Operation of Triangular Fuzzy Number......Page 150
5.3.3 Operation of General Fuzzy Numbers......Page 152
5.3.4 Approximation of Triangular Fuzzy Number......Page 155
5.4.1 Trapezoidal Fuzzy Number......Page 156
5.4.2 Operations of Trapezoidal Fuzzy Number......Page 157
5.4.3 Bell Shape Fuzzy Number......Page 158
Summary......Page 160
Exercises......Page 161
6.1.1 Function with Fuzzy Constraint......Page 164
6.1.2 Propagation of Fuzziness by Crisp Function......Page 165
6.1.3 Fuzzifying Function of Crisp Variable......Page 166
6.2.1 Maximizing and Minimizing Set......Page 169
6.2.2 Maximum Value of Crisp Function......Page 170
6.3.1 Integration......Page 174
6.3.2 Differentiation......Page 177
Summary......Page 179
Exercises......Page 180
7.1.1 Probability Theory......Page 182
7.1.2 Possibility Distribution......Page 183
7.1.3 Comparison of Probability and Possibility......Page 184
7.2 Fuzzy Event......Page 185
7.2.1 Crisp Probability of Fuzzy Event......Page 186
7.2.2 Fuzzy Probability of Fuzzy Event......Page 188
7.3.1 Uncertainty Level of Element......Page 190
7.3.2 Fuzziness of Fuzzy Set......Page 191
7.4.1 Definition......Page 192
7.4.2 Measure using Entropy......Page 194
7.4.3 Measure using Metric Distance......Page 197
Summary......Page 200
Exercises......Page 201
8.1.1 Proposition Logic......Page 204
8.1.2 Logic Function......Page 207
8.1.3 Tautology and Inference Rule......Page 209
8.1.4 Predicate Logic......Page 210
8.1.5 Quantifier......Page 211
8.2.1 Fuzzy Expression......Page 212
8.2.2 Operators in Fuzzy Expression......Page 213
8.2.3 Some Examples of Fuzzy Logic Operations......Page 214
8.3.1 Definition of Linguistic Variable......Page 215
8.3.3 Fuzzy Modifier......Page 216
8.4.1 Fuzzy Truth Values......Page 217
8.4.2 Examples of Fuzzy Truth Qualifier......Page 219
8.5.2 Representation of Fuzzy Predicate by Fuzzy Relation......Page 221
8.5.3 Representation of Fuzzy Rule......Page 222
Summary......Page 224
Exercises......Page 226
9.1.1 Extension Principle and Composition......Page 228
9.1.3 Composition of Fuzzy Relations......Page 229
9.1.4 Example of Fuzzy Composition......Page 230
9.2.1 Fuzzy if-then Rules......Page 232
9.2.3 Example of Fuzzy Implications......Page 233
9.3.1 Decomposition of Rule Base......Page 235
9.3.3 Compositional Rule of Inference......Page 237
9.3.4 Fuzzy Inference with Rule Base......Page 239
9.4 Inference Methods......Page 247
9.4.1 Mamdani Method......Page 249
9.4.2 Larsen Method......Page 252
9.4.3 Tsukamoto Method......Page 255
9.4.4 TSK Method......Page 256
Summary......Page 258
Exercises......Page 261
10.1.1 Advantage of Fuzzy Logic Controller......Page 264
10.1.2 Configuration of Fuzzy Logic Controller......Page 265
10.2 Fuzzification Interface Component......Page 266
10.3.1 Data Base......Page 268
10.3.2 Rule Base......Page 273
10.4 Inference (Decision Making Logic)......Page 276
10.4.2 Larsen Method......Page 277
10.4.3 Tsukamoto Method......Page 278
10.5.1 Mean of Maximum Method (MOM)......Page 280
10.5.2 Center of Area Method (COA)......Page 281
10.5.4 Lookup Table......Page 282
10.6 Design Procedure of Fuzzy Logic Controller......Page 283
10.7 Application Example of FLC Design......Page 284
10.8 Fuzzy Expert Systems......Page 288
10.8.1 Fuzzification Interface......Page 289
10.8.5 Scheduler......Page 290
Summary......Page 291
Exercises......Page 293
11.1.1 Basic Concepts of Neural Networks......Page 296
11.1.2 Learning Algorithms......Page 297
11.1.3 Multilayer Perceptrons and Error Backpropagation Learning......Page 298
11.2.1 Modifying Fuzzy Systems with Supervised Neural Network Learning......Page 301
11.2.2 Building Neural Networks using Fuzzy Systems......Page 306
11.2.3 Making Membership Functions with Neural Networks......Page 309
11.2.5 Concatenating Neural Networks and Fuzzy Systems......Page 312
Summary......Page 317
Exercises......Page 319
12.1.1 General Structure of Genetic Algorithms......Page 320
12.1.2 Evolution of Genetic Algorithms......Page 321
12.2.1 Identifying Fuzzy Systems with Genetic Algorithms......Page 325
12.2.2 Controlling Parameters of Genetic Algorithms with Fuzzy Systems......Page 331
Summary......Page 334
Exercises......Page 335
Bibliography......Page 336
Index......Page 344
Back Cover......Page 348