# Fuzzy Mathematics: An Introduction for Engineers and Scientists

Springer Science & Business Media, Sep 7, 2001 - Mathematics - 314 pages
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 Copyright