Table Of Content(cid:2)
AModernIntroductiontoFuzzyMathematics
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A Modern Introduction to Fuzzy
Mathematics
Apostolos Syropoulos
Theophanes Grammenos
NewYork2020
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Thiseditionfirstpublished2020
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LibraryofCongressCataloging-in-PublicationData
Names:Syropoulos,Apostolos,author.
Title:Amodernintroductiontofuzzymathematics/ApostolosSyropoulos,
TheophanesGrammenos.
Description:Firstedition.|NewYork:Wiley,2020.|Includes
bibliographicalreferencesandindex.
Identifiers:LCCN2020004300(print)|LCCN2020004301(ebook)|ISBN
9781119445289(cloth)|ISBN9781119445302(adobepdf)|ISBN
9781119445296(epub)
Subjects:LCSH:Fuzzymathematics.
Classification:LCCQA248.5.S972020(print)|LCCQA248.5(ebook)|DDC
511.3/13—dc23
LCrecordavailableathttps://lccn.loc.gov/2020004300
LCebookrecordavailableathttps://lccn.loc.gov/2020004301
CoverDesign:Wiley
CoverImage:CourtesyofApostolosSyropoulos
Setin9.5/12.5ptSTIXTwoTextbySPiGlobal,Chennai,India
PrintedintheUnitedStatesofAmerica
10987654321
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TomysonDemetrios-Georgios, TomywifeAndromacheforher
toKoulaandLinda. endurance,tolerance,andcare.
ApostolosSyropoulos TheophanesGrammenos
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vii
Contents
Preface xiii
Acknowledgments xvii
1 Introduction 1
1.1 WhatIsVagueness? 1
1.2 Vagueness,Ambiguity,Uncertainty,etc. 4
1.3 VaguenessandFuzzyMathematics 6
Exercises 8
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2 FuzzySetsandTheirOperations 11
2.1 AlgebrasofTruthValues 11
2.1.1 Posets 12
2.1.2 Lattices 13
2.1.3 Frames 14
2.2 Zadeh’sFuzzySets 14
2.3 𝛼-CutsofFuzzySets 19
2.4 Interval-valuedandType2FuzzySets 21
2.5 TriangularNormsandConorms 24
2.6 L-fuzzySets 26
2.7 “Intuitionistic”FuzzySetsandTheirExtensions 27
2.8 TheExtensionPrinciple 32
2.9* Boolean-ValuedSets 34
2.10* AxiomaticFuzzySetTheory 36
Exercises 37
3 FuzzyNumbersandTheirArithmetic 39
3.1 FuzzyNumbers 39
3.1.1 TriangularFuzzyNumbers 40
3.1.2 TrapezoidalFuzzyNumbers 41
3.1.3 GaussianFuzzyNumbers 42
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viii Contents
3.1.4 QuadraticFuzzyNumbers 43
3.1.5 ExponentialFuzzyNumbers 44
3.1.6 L–RFuzzyNumbers 44
3.1.7 GeneralizedFuzzyNumbers 46
3.2 ArithmeticofFuzzyNumbers 46
3.2.1 IntervalArithmetic 47
3.2.2 IntervalArithmeticand𝛼-Cuts 47
3.2.3 FuzzyArithmeticandtheExtensionPrinciple 48
3.2.4 FuzzyArithmeticofTriangularFuzzyNumbers 49
3.2.5 FuzzyArithmeticofGeneralizedFuzzyNumbers 49
3.2.6 ComparingFuzzyNumbers 51
3.3 LinguisticVariables 54
3.4 FuzzyEquations 55
3.4.1 SolvingtheFuzzyEquationA⋅X+B=C 55
3.4.1.1 TheClassicalMethod 55
3.4.1.2 TheExtensionPrincipleMethod 56
3.4.1.3 The𝛼-CutsMethod 58
3.4.2 SolvingtheFuzzyEquationA⋅X2+B⋅X+C=D 58
3.4.2.1 TheClassicalMethod 59
3.4.2.2 TheExtensionPrincipleMethod 59
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3.4.2.3 The𝛼-CutsMethod 60
3.5 FuzzyInequalities 60
3.6 ConstructingFuzzyNumbers 60
3.7 ApplicationsofFuzzyNumbers 63
3.7.1 SimulationoftheHumanGlucoseMetabolism 63
3.7.2 EstimationofanOngoingProject’sCompletionTime 66
3.7.2.1 ModelofaProject 67
Exercises 68
4 FuzzyRelations 71
4.1 CrispRelations 71
4.1.1 PropertiesofRelations 72
4.1.2 NewRelationsfromOldOnes 72
4.1.3 RepresentingRelationsUsingMatrices 73
4.1.4 RepresentingRelationsUsingDirectedGraphs 73
4.1.5 TransitiveClosureofaRelation 74
4.1.6 EquivalenceRelations 75
4.2 FuzzyRelations 75
4.3 CartesianProduct,Projections,andCylindricalExtension 77
4.3.1 CartesianProduct 77
4.3.2 ProjectionofFuzzyRelations 77
4.3.3 CylindricalExtension 78
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Contents ix
4.4 NewFuzzyRelationsfromOldOnes 78
4.5 FuzzyBinaryRelationsonaSet 81
4.5.1 TransitiveClosure 82
4.5.2 SimilarityRelations 83
4.5.3 ProximityRelations 83
4.6 FuzzyOrders 87
4.7 ElementsofFuzzyGraphTheory 89
4.7.1 GraphsandHypergraphs 89
4.7.2 FuzzyGraphs 91
4.7.2.1 PathsandConnectedness 92
4.7.2.2 BridgesandCutVertices 93
4.7.2.3 FuzzyTreesandFuzzyForests 93
4.7.3 FuzzyHypergraphs 96
4.8* FuzzyCategoryTheory 98
4.8.1 CommutativeDiagrams 99
4.8.2 CategoriesofFuzzyStructures 99
4.8.3 EmbeddingFuzzyCategoriestoChuCategories 101
4.8.4 FuzzyCategories 103
4.9* FuzzyVectors 105
4.10 Applications 107
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Exercises 109
5 PossibilityTheory 111
5.1 FuzzyRestrictionsandPossibilityTheory 111
5.2 PossibilityandNecessityMeasures 113
5.3 PossibilityTheory 115
5.4 PossibilityTheoryandProbabilityTheory 118
5.5 AnUnexpectedApplicationofPossibilityTheory 122
Exercises 128
6 FuzzyStatistics 129
6.1 RandomVariables 129
6.2 FuzzyRandomVariables 132
6.3 PointEstimation 136
6.3.1 TheUnbiasedEstimator 137
6.3.2 TheConsistentEstimator 138
6.3.3 TheMaximumLikelihoodEstimator 139
6.4 FuzzyPointEstimation 140
6.5 IntervalEstimation 141
6.6 IntervalEstimationforFuzzyData 143
6.7 HypothesisTesting 144
6.8 FuzzyHypothesisTesting 146
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x Contents
6.9 StatisticalRegression 148
6.10 FuzzyRegression 151
Exercises 153
7 FuzzyLogics 155
7.1 MathematicalLogic 155
7.2 Many-ValuedLogics 161
7.3 OnFuzzyLogics 166
7.4 Hájek’sBasicMany-ValuedLogic 168
7.5 ŁukasiewiczFuzzyLogic 171
7.6 ProductFuzzyLogic 172
7.7 GödelFuzzyLogic 174
7.8 First-OrderFuzzyLogics 176
7.9 FuzzyQuantifiers 178
7.10 ApproximateReasoning 179
7.11 Application:FuzzyExpertSystems 182
7.11.1 Fuzzification 184
7.11.2 EvaluationofRules 184
7.11.3 Defuzzification 185
7.12* ALogicofVagueness 187
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Exercises 189
8 FuzzyComputation 191
8.1 Automata,Grammars,andMachines 191
8.2 FuzzyLanguagesandGrammars 196
8.3 FuzzyAutomata 200
8.4 FuzzyTuringMachines 205
8.5 OtherFuzzyModelsofComputation 209
9 FuzzyAbstractAlgebra 215
9.1 Groups,Rings,andFields 215
9.2 FuzzyGroups 219
9.3 AbelianFuzzySubgroups 224
9.4 FuzzyRingsandFuzzyFields 227
9.5 FuzzyVectorSpaces 229
9.6 FuzzyNormedSpaces 230
9.7 FuzzyLieAlgebras 231
Exercises 233
10 FuzzyTopology 235
10.1 MetricandTopologicalSpaces 235
10.2 FuzzyMetricSpaces 240
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Contents xi
10.3 FuzzyTopologicalSpaces 245
10.4 FuzzyProductSpaces 248
10.5 FuzzySeparation 250
10.5.1 Separation 256
10.6 FuzzyNets 256
10.7 FuzzyCompactness 257
10.8 FuzzyConnectedness 258
10.9 SmoothFuzzyTopologicalSpaces 258
10.10 FuzzyBanachandFuzzyHilbertSpaces 260
10.11* FuzzyTopologicalSystems 263
Exercises 267
11 FuzzyGeometry 269
11.1 FuzzyPointsandFuzzyDistance 269
11.2 FuzzyLinesandTheirProperties 272
11.3 FuzzyCircles 276
11.4 RegularFuzzyPolygons 280
11.5 ApplicationsinTheoreticalPhysics 285
Exercises 287
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12 FuzzyCalculus 289
12.1 FuzzyFunctions 289
12.2 IntegralsofFuzzyFunctions 294
12.3 DerivativesofFuzzyFunctions 298
12.4 FuzzyLimitsofSequencesandFunctions 300
12.4.1 FuzzyOrdinaryDifferentialEquations 304
12.4.2 FuzzyPartialDifferentialEquations 310
Exercises 313
A FuzzyApproximation 315
A.1 WeierstrassandStone–WeierstrassApproximationTheorems 315
A.2 WeierstrassandStone–WeierstrassFuzzyAnalogs 316
B ChaosandVagueness 319
B.1 ChaosTheoryinaNutshell 319
B.2 FuzzyChaos 322
B.3 FuzzyFractals 324
WorksCited 327
SubjectIndex 349
AuthorIndex 361
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