Fuzzy Mathematics: An Introduction for Engineers and Scientists

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Springer Science & Business Media, Sep 7, 2001 - Mathematics - 314 pages
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In the mid-1960's I had the pleasure of attending a talk by Lotfi Zadeh at which he presented some of his basic (and at the time, recent) work on fuzzy sets. Lotfi's algebra of fuzzy subsets of a set struck me as very nice; in fact, as a graduate student in the mid-1950's, I had suggested similar ideas about continuous-truth-valued propositional calculus (inffor "and", sup for "or") to my advisor, but he didn't go for it (and in fact, confused it with the foundations of probability theory), so I ended up writing a thesis in a more conventional area of mathematics (differential algebra). I especially enjoyed Lotfi's discussion of fuzzy convexity; I remember talking to him about possible ways of extending this work, but I didn't pursue this at the time. I have elsewhere told the story of how, when I saw C. L. Chang's 1968 paper on fuzzy topological spaces, I was impelled to try my hand at fuzzi fying algebra. This led to my 1971 paper "Fuzzy groups", which became the starting point of an entire literature on fuzzy algebraic structures. In 1974 King-Sun Fu invited me to speak at a U. S. -Japan seminar on Fuzzy Sets and their Applications, which was to be held that summer in Berkeley.
 

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Contents

FUZZY SUBSETS
1
11 Fuzzy Relations
6
12 Operations on Fuzzy Relations
8
13 Reflexivity Symmetry and Transitivity
11
14 Pattern Classification Based on Fuzzy Relations
12
15 Advanced Topics on Fuzzy Relations
16
16 References
20
FUZZY GRAPHS
21
410 Fuzzy Digital Convexity
131
411 On Connectivity Properties of Grayscale Pictures
133
412 References
135
FUZZY GEOMETRY
137
53 The Height Width and Diameter of a Fuzzy Subset
147
54 Distances Between Fuzzy Subsets
152
55 Fuzzy Rectangles
155
56 A Fuzzy Medial Axis Transformation Based on Fuzzy Disks
158

21 Paths and Connectedness
22
22 Clusters
24
23 Cluster Analysis and Modeling of Information Networks
29
24 Connectivity in Fuzzy Graphs
32
25 Application to Cluster Analysis
39
26 Operations on Fuzzy Graphs
44
27 Fuzzy Intersection Equations
52
28 Fuzzy Graphs in Database Theory
58
29 References
61
FUZZY TOPOLOGICAL SPACES
67
32 Metric Spaces and Normed Linear Spaces
74
33 Fuzzy Topological Spaces
79
34 Sequences of Fuzzy Subsets
81
35 FContinuous Functions
82
36 Compact Fuzzy Spaces
84
37 Iterated Fuzzy Subset Systems
85
38 Chaotic Iterations of Fuzzy Subsets
95
39 Starshaped Fuzzy Subsets
99
310 References
102
FUZZY DIGITAL TOPOLOGY
115
43 Fuzzy Connectedness
116
44 Fuzzy Components
118
45 Fuzzy Surroundedness
123
46 Components Holes and Surroundedness
124
47 Convexity
127
48 The Sup Projection
128
57 Fuzzy Triangles
163
58 Degree of Adjacency or Surroundedness
166
59 Image Enhancement and Thresholding Using Fuzzy Compactness
181
Points and Lines
189
Circles and Polygons
197
512 Fuzzy Plane Projective Geometry
204
513 A Modified Hausdorff Distance Between Fuzzy Subsets
207
514 References
214
FUZZY ABSTRACT ALGEBRA
219
62 Fuzzy Substructures of Algebraic Structures
233
63 Fuzzy Submonoids and Automata Theory
238
64 Fuzzy Subgroups Pattern Recognition and Coding Theory
240
65 Free Fuzzy Monoids and Coding Theory
245
66 Formal Power Series Regular Fuzzy Languages and Fuzzy Automata
252
67 Nonlinear Systems of Equations of Fuzzy Singletons
266
68 Localized Fuzzy Subrings
272
69 Local Examination of Fuzzy Intersection Equations
276
610 More on Coding Theory
281
611 Other Applications
286
612 References
287
LIST OF FIGURES
291
LIST OF TABLES
293
LIST OF SYMBOLS
295
INDEX
303
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