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.

With numerous examples Presentation of concepts of fuzzy mathematics with applications to engineering, computer science, and mathematics Includes supplementary material: sn.pub/extras

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Dr. John N. Mordeson is Professor Emeritus of Mathematics at Creighton University. He received his B.S., M.S., and Ph.D. from Iowa State University. He is a Member of Phi Kappa Phi. He is a President of the Society for Mathematics of Uncertainty. He has published fteen books and two hundred journal articles. He is on the editorial board of numerous journals. He has served as an external examiner of Ph.D. candidates from India, South Africa, Bulgaria, and Pakistan. He has refereed for numerous journals and granting agencies. He is particularly interested in applying mathematics of uncertainty to combat the problem of human traf cking. Dr. Sunil Mathew is currently a Faculty Member in the Department of Mathematics, NIT Calicut, India. He has acquired his masters from St. Joseph's College Devagiri, Calicut, and Ph.D. from National Institute of Technology Calicut in the area of Fuzzy Graph Theory. He has published more than seventy- ve research papers and written two books. He is a Member of several academic bodies and associations. He is editor and reviewer of several international journals. He has an experience of twenty years in teaching and research. His current research topics include fuzzy graph theory, bio-computational modeling, graph theory, fractal geometry, and chaos. Dr. Davender S. Malik is a Professor of Mathematics at Creighton University. He received his Ph.D. from Ohio University and has published more than fty- ve papers and eighteen books on abstract algebra, applied mathematics, graph theory, fuzzy automata theory and languages, fuzzy logic and its applications, programming, data structures, and discrete mathematics.

Inhalt
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.