Mathematics of Fuzzy Sets and Fuzzy Logic

280 Pages·2013·3.12 MB·English

by  Barnabas Bede (auth.)

Studies in Fuzziness and Soft Computing 295
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Prof.JanuszKacprzyk
SystemsResearchInstitute
PolishAcademyofSciences
ul.Newelska6
01-447Warsaw
Poland
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Forfurthervolumes:
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Barnabas Bede
Mathematics of Fuzzy Sets
and Fuzzy Logic
ABC
Author
BarnabasBede
DepartmentofMathematics
DigiPenInstituteofTechnology
Redmond,WA
USA
ISSN1434-9922 e-ISSN1860-0808
ISBN978-3-642-35220-1 e-ISBN978-3-642-35221-8
DOI10.1007/978-3-642-35221-8
SpringerHeidelbergNewYorkDordrechtLondon
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(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Dedicated to my daughters: Fruzsina and Tekla
Preface
The present manuscript is intended to be a textbook that could serve both
undergraduate and graduate students when studying Fuzzy Set Theory and
Applications. It is also intended to deepen the research into some existing
directions and to investigate some new research directions. The book tries
to develop a systematic, Mathematically-based introduction into the theory
and applications of fuzzy sets and fuzzy logic. In this way the author tries
to cover a gap in the literature. Also, the book can be an introduction into
Fuzzy Sets and Systems for researchers who are interested in the topic of
fuzzy sets in all areas of Mathematics, Computer Science and Engineering,
or simply interested Mathematicians, Engineers and students in these areas.
The book starts from the basic theory and gets the reader to a level very
close to the current researchtopics in Fuzzy Sets.
The basis ofthe book is the author’sclass notes for Fuzzy Sets and Fuzzy
Logic, class taught at DigiPen Institute of Technology.
Another goal that the author had when writing the presentbook is try to
see where Fuzzy Sets and Fuzzy Logic as a discipline, can be connected to
other areas of Mathematics. The manuscript tries to show that Fuzzy Sets
is an independent discipline having huge overlaps on one side with Analysis
andApproximationTheoryandontheothersidewithLogicandSetTheory.
The approach of the present work leans toward Mathematical Analysis and
Approximation.
I am really thankful to my wife Emese, who providedverystrong support
formeinwritingthisbook.Also,the authoris thankfultohis colleagues,es-
pecially Sorin G. Gal, L. Stefanini, L.C. Barros, I.J. Rudas and many others
forcommentsthatimprovedthemanuscript.Theauthorwouldliketoexpress
VIII Preface
histhankstoallthestudentswhocontributedwithcommentsthatimproved
the manuscript, especially Kia McDowel, Matt Peterson, Chris Barrett, and
many others.
Barnabas Bede
Redmond, WA
Contents
1 Fuzzy Sets 1
1.1 Classical Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 The Basic Connectives . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Intersection . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.3 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.4 Complementation. . . . . . . . . . . . . . . . . . . . . 6
1.4 Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Fuzzy Set-Theoretic Operations 13
2.1 Negation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Triangular Norms and Conorms . . . . . . . . . . . . . . . . . 16
2.3 Archimedean t-Norms and t-Conorms . . . . . . . . . . . . . 19
2.4 Fuzzy Implications . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Fuzzy Equivalence . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Fuzzy Relations 33
3.1 Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Max-Min Composition . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Min-Max Composition . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Min → Composition . . . . . . . . . . . . . . . . . . . . . . . 38
X Contents
3.5 Fuzzy Relational Equations with Max-Min and Min →
Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Max-t-Norm Composition . . . . . . . . . . . . . . . . . . . . 43
3.7 Min → Composition . . . . . . . . . . . . . . . . . . . . . . 44
T
3.8 Fuzzy Relational Equations with Max-t-Norm and Min →
T
Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Fuzzy Numbers 51
4.1 Definition of Fuzzy Numbers . . . . . . . . . . . . . . . . . . 51
4.2 Characterization Theorems for Fuzzy Numbers . . . . . . . . 56
4.3 L-R Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5 Fuzzy Arithmetic 65
5.1 Zadeh’s Extension Principle . . . . . . . . . . . . . . . . . . . 65
5.2 The Sum and Scalar Multiplication . . . . . . . . . . . . . . . 69
5.3 The Product of Two Fuzzy Numbers . . . . . . . . . . . . . . 70
5.4 Difference of Fuzzy Numbers . . . . . . . . . . . . . . . . . . 73
5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6 Fuzzy Inference 79
6.1 Linguistic Variables. . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 Fuzzy Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3 Fuzzy Rule Base . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.4 Fuzzy Inference . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.5 The Interpolation Property of a Fuzzy Inference System . . . 88
6.6 Example of a Fuzzy Inference System . . . . . . . . . . . . . 98
6.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7 Single Input Single Output Fuzzy Systems 105
7.1 Structure of a SISO Fuzzy System . . . . . . . . . . . . . . . 106
7.1.1 Fuzzification . . . . . . . . . . . . . . . . . . . . . . . 106
7.1.2 Fuzzy Rule Base . . . . . . . . . . . . . . . . . . . . . 107
7.1.3 Fuzzy Inference . . . . . . . . . . . . . . . . . . . . . . 107
7.1.4 Defuzzification . . . . . . . . . . . . . . . . . . . . . . 107
7.2 Fuzzy Inference and Rule Base for a SISO Fuzzy System . . 108
7.3 Approximation Properties of SISO Fuzzy Systems . . . . . . 113
7.3.1 ApproximationbyMamdani,Larsenandt-NormBased
SISO Fuzzy Systems . . . . . . . . . . . . . . . . . . . 114
7.3.2 Approximation by SISO Fuzzy System of Go¨del and
G¨odel Residual Types . . . . . . . . . . . . . . . . . . 118
7.4 Takagi-Sugeno Fuzzy System . . . . . . . . . . . . . . . . . . 123
7.5 Approximation Properties of Takagi-Sugeno Fuzzy Systems . 125
7.6 Fuzzy Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Contents XI
7.7 Example of a Fuzzy Controller . . . . . . . . . . . . . . . . . 130
7.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8 Fuzzy Analysis 137
8.1 Metric Spaces of Fuzzy Numbers . . . . . . . . . . . . . . . . 137
8.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.4 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.5 Norm of a Fuzzy Number . . . . . . . . . . . . . . . . . . . . 150
8.6 Embedding Theorem for Fuzzy Numbers . . . . . . . . . . . . 151
8.7 Fuzzy Numbers with Continuous Endpoints of the
Level Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.8 Integration of Fuzzy-Number-Valued Functions . . . . . . . . 154
8.9 Differentiability of Fuzzy-Number-Valued Functions . . . . . 157
8.9.1 Hukuhara Differentiability . . . . . . . . . . . . . . . . 157
8.9.2 Generalized Differentiabilities . . . . . . . . . . . . . . 159
8.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
9 Fuzzy Differential Equations 171
9.1 FDEs under Hukuhara Differentiability . . . . . . . . . . . . 171
9.2 The Interpretation Based on Zadeh’s Extension Principle . . 176
9.3 Fuzzy Differential Equations . . . . . . . . . . . . . . . . . . . 179
9.3.1 Existence and Uniqueness of Two Solutions . . . . . . 179
9.3.2 CharacterizationResults . . . . . . . . . . . . . . . . . 184
9.3.3 Examples of Fuzzy Differential Equations under
Strongly Generalized Differentiability . . . . . . . . . 186
9.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
10 Extensions of Fuzzy Set Theory 193
10.1 Lattice Valued Fuzzy Sets (L-Fuzzy Sets) . . . . . . . . . . . 193
10.2 Intuitionistic Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . 194
10.3 Interval Type II Fuzzy Sets . . . . . . . . . . . . . . . . . . . 195
10.4 Fuzzy Sets of Type 2 . . . . . . . . . . . . . . . . . . . . . . . 196
10.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
11 Possibility Theory 201
11.1 Fuzzy Sets and Possibility Distributions . . . . . . . . . . . . 201
11.2 Fuzzy Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 202
11.3 Possibility Measures . . . . . . . . . . . . . . . . . . . . . . . 205
11.4 Fuzzy Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 206
11.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
12 Fuzzy Clustering 213
12.1 Classical k-Means Clustering . . . . . . . . . . . . . . . . . . 213
12.2 Fuzzy c-Means . . . . . . . . . . . . . . . . . . . . . . . . . . 215
XII Contents
12.3 Continuous Fuzzy c-Means . . . . . . . . . . . . . . . . . . . 217
12.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
13 Fuzzy Transform 221
13.1 Definition of Fuzzy Transforms . . . . . . . . . . . . . . . . . 221
13.2 Error Estimates for F-Transforms Based on Fuzzy
Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
13.3 B-Splines Based F-Transform . . . . . . . . . . . . . . . . . . 228
13.4 Shepard Kernels Based F-Transform . . . . . . . . . . . . . . 231
13.5 Korovkin Type Theorems for the F-Transform . . . . . . . . 236
13.6 F-Transform with Bernstein Basis Polynomials . . . . . . . . 239
13.7 F-Transform of Favard-Sz´asz-MirakjanType. . . . . . . . . . 242
13.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
14 Artificial Neural Networks and Neuro-Fuzzy Systems 247
14.1 Artificial Neuron . . . . . . . . . . . . . . . . . . . . . . . . . 247
14.2 Feed-Forward Neural Network . . . . . . . . . . . . . . . . . . 248
14.3 Learning of a Neural Network . . . . . . . . . . . . . . . . . . 251
14.4 Approximation Properties of Neural Networks . . . . . . . . . 253
14.5 Adaptive Network Based Fuzzy Inference System (ANFIS). . 254
14.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
References 259
Appendix A Mathematical Prerequisites 267
A.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
A.2 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
A.3 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
A.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
A.5 Modulus of Continuity . . . . . . . . . . . . . . . . . . . . . . 272
A.6 Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Index 275