vi CONTENTS
6
Fuzzy Relations and Fuzzy Graphs 71
6.1
Fuzzy Relations on Sets and Fuzzy Sets
71
6.1.1
Compositions of Fuzzy Relations 76
6.1.2
Properties of the Min-Max Composition 79
6.2
Fuzzy Graphs
83
6.3
Special Fuzzy Relations
86
7
Fuzzy Analysis
93
7.1
Fuzzy Functions on Fuzzy Sets
93
7.2
Extrema of Fuzzy Functions
95
7.3
Integration of Fuzzy Functions
99
7.3.1
Integration of a Fuzzy Function over a Crisp Interval 100
7.3.2
Integration of a (Crisp) Real-Valued Function over a Fuzzy
Interval
103
7.4
Fuzzy Differentiation
107
8
Uncertainty Modeling
111
8.1
Application-oriented Modeling of Uncertainty
111
8.1.1 Causes of Uncertainty 114
8.1.2
Type of Available Information 117
8.1.3
Uncertainty Methods 118
8.1.4
Uncertainty Theories as Transformers of Information 119
8.1.5
Matching Uncertainty Theory and Uncertain Phenomena 120
8.2
Possibility Theory 122
8.2.1
Fuzzy Sets and Possibility Distributions
122
8.2.2
Possibility and Necessity Measures 126
8.3
Probability of Fuzzy Events 129
8.3.1
Probability of a Fuzzy Event as a Scalar 129
8.3.2
Probability of a Fuzzy Event as a Fuzzy Set
131
8.4
Possibility vs. Probability 133
Part II: Applications of Fuzzy Set Theory 139
9
Fuzzy Logic and Approximate Reasoning
141
9.1 Linguistic Variables 141
9.2
Fuzzy Logic 149
9.2.1
Classical Logics Revisited 149
9.2.2
Linguistic Truth Tables
153
9.3
Approximate and Plausible Reasoning 156
9.4
Fuzzy Languages 160
9.5
Support Logic Programming and Fril
169
9.5.1
Introduction 169
9.5.2
Fril Rules 170
9.5.3
Inference Methods in Fril
172
9.5.4
Fril Inference for a Single Rule 175
9.5.5
Multiple Rule Case 176
9.5.6
Interval and Point Semantic Unification
177
9.5.7
Least Prejudiced Distribution and Learning 179
9.5.8
Applications of Fril
181
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