## Fuzzy Mathematics: An Introduction for Engineers and ScientistsIn 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 |

287 | |

LIST OF FIGURES | 291 |

LIST OF TABLES | 293 |

LIST OF SYMBOLS | 295 |

303 | |

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### Common terms and phrases

A A(Q affine variety algebraic algorithm border called a fuzzy clusters commutative ring compact component computing connected Consider contains contraction map crisp Define the fuzzy Definition digital topology edge element equivalent Example exists finite free monoid free semigroup fuzzy disk fuzzy forest fuzzy graph fuzzy ideal fuzzy line fuzzy numbers fuzzy points fuzzy relation Fuzzy Sets fuzzy singletons fuzzy subgroup fuzzy subset fuzzy topological space fuzzy topology geometry graph G grey level Hankel matrix Hausdorff distance Hence identity and let Lemma Let G level sets line segment matrix maximal medial axis metric space neighborhood nonempty partial fuzzy subgraph path Pattern Recognition perimeter pixels plane polynomial ring prime ideal Proof properties Proposition real numbers reflexive region respectively ring with identity Rosenfeld sequence Sets and Systems submonoid subsemigroup submonoid Suppose surroundedness t-cuts Theorem theory threshold topological space vector vertex vertices