1 Fuzzy Subsets -- 1.1 Fuzzy Relations -- 1.2 Operations on Fuzzy Relations -- 1.3 Reflexivity, Symmetry and Transitivity -- 1.4 Pattern Classification Based on Fuzzy Relations -- 1.5 Advanced Topics on Fuzzy Relations -- 1.6 References -- 2 Fuzzy Graphs -- 2.1 Paths and Connectedness -- 2.2 Clusters -- 2.3 Cluster Analysis and Modeling of Information Networks -- 2.4 Connectivity in Fuzzy Graphs -- 2.5 Application to Cluster Analysis -- 2.6 Operations on Fuzzy Graphs -- 2.7 Fuzzy Intersection Equations -- 2.8 Fuzzy Graphs in Database Theory -- 2.9 References -- 3 Fuzzy Topological Spaces -- 3.1 Topological Spaces -- 3.2 Metric Spaces and Normed Linear Spaces -- 3.3 Fuzzy Topological Spaces -- 3.4 Sequences of Fuzzy Subsets -- 3.5 F-Continuous Functions -- 3.6 Compact Fuzzy Spaces -- 3.7 Iterated Fuzzy Subset Systems -- 3.8 Chaotic Iterations of Fuzzy Subsets -- 3.9 Starshaped Fuzzy Subsets -- 3.10 References -- 4 Fuzzy Digital Topology -- 4.1 Introduction -- 4.2 Crisp Digital Topology -- 4.3 Fuzzy Connectedness -- 4.4 Fuzzy Components -- 4.5 Fuzzy Surroundedness -- 4.6 Components, Holes, and Surroundedness -- 4.7 Convexity -- 4.8 The Sup Projection -- 4.9 The Integral Projection -- 4.10 Fuzzy Digital Convexity -- 4.11 On Connectivity Properties of Grayscale Pictures -- 4.12 References -- 5 Fuzzy Geometry -- 5.1 Introduction -- 5.2 The Area and Perimeter of a Fuzzy Subset -- 5.3 The Height, Width and Diameter of a Fuzzy Subset -- 5.4 Distances Between Fuzzy Subsets -- 5.5 Fuzzy Rectangles -- 5.6 A Fuzzy Medial Axis Transformation Based on Fuzzy Disks -- 5.7 Fuzzy Triangles -- 5.8 Degree of Adjacency or Surroundedness -- 5.9 Image Enhancement and Thresholding Using Fuzzy Compactness -- 5.10 Fuzzy Plane Geometry: Points and Lines -- 5.11 Fuzzy Plane Geometry: Circles and Polygons -- 5.12 Fuzzy Plane Projective Geometry -- 5.13 A Modified Hausdorff Distance Between Fuzzy Subsets. -- 5.14 References -- 6 Fuzzy Abstract Algebra -- 6.1 Crisp Algebraic Structures -- 6.2 Fuzzy Substructures of Algebraic Structures -- 6.3 Fuzzy Submonoids and Automata Theory -- 6.4 Fuzzy Subgroups, Pattern Recognition and Coding Theory -- 6.5 Free Fuzzy Monoids and Coding Theory -- 6.6 Formal Power Series, Regular Fuzzy Languages, and Fuzzy Automata -- 6.7 Nonlinear Systems of Equations of Fuzzy Singletons -- 6.8 Localized Fuzzy Subrings -- 6.9 Local Examination of Fuzzy Intersection Equations -- 6.10 More on Coding Theory -- 6.11 Other Applications -- 6.12 References -- List of Figures -- List of Tables -- List of Symbols.

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.