## Metric Spaces of Fuzzy Sets: Theory and ApplicationsThe primary aim of the book is to provide a systematic development of the theory of metric spaces of normal, upper semicontinuous fuzzy convex fuzzy sets with compact support sets, mainly on the base space ℜn. An additional aim is to sketch selected applications in which these metric space results and methods are essential for a thorough mathematical analysis.This book is distinctly mathematical in its orientation and style, in contrast with many of the other books now available on fuzzy sets, which, although all making use of mathematical formalism to some extent, are essentially motivated by and oriented towards more immediate applications and related practical issues. The reader is assumed to have some previous undergraduate level acquaintance with metric spaces and elementary functional analysis. |

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### Contents

Preface | 1 |

Spaces of Subsets of | 7 |

Compact Convex Subsets of 9t | 13 |

Set Valued Mappings | 21 |

Crisp Generalizations | 33 |

Metrics on n | 51 |

Compactness Criteria | 71 |

Generalizations | 81 |

Fuzzy Random Variables | 109 |

Computational Methods | 115 |

Fuzzy Differential Equations | 129 |

Optimization Under Uncertainty | 137 |

Fuzzy Iterations and Image Processing | 143 |

Appendix on Metric Spaces | 155 |

Bibliography | 161 |

171 | |

### Other editions - View all

Metric Spaces of Fuzzy Sets: Theory and Applications Phil Diamond,Peter Kloeden Limited preview - 1994 |

### Common terms and phrases

Aumann integrable Banach space base space Blasi differentiable Bobylev Cauchy sequence closed subset compact convex subsets compact subset complete metric space cone constraints continuous functions convergence convex sets crisp defined definition denote differentiable at t0 equileftcontinuous equivalent Example follows fuzzy convex fuzzy differential equation fuzzy random variables fuzzy set valued fuzzy star shaped grey level Hausdorff distance Hausdorff metric Hence Hukuhara derivative Hukuhara difference Hukuhara differentiable IFZS Kaleva Lemma Let F level set mappings levelsetwise Lipschitz continuous lower semicontinuous nondecreasing sequence nonempty compact convex nonempty compact subset nonempty subset p-Blaschke properties Proposition 2.4.3 Proposition 6.1.7 Puri and Ralescu random variables satisfies send(u sendograph set u G set valued mapping solution star shaped sets Steiner centroid subset of 5R subspace support function supremum Theorem topological entropy topology triangle inequality triangular fuzzy numbers u,v G uniformly support-bounded unique upper semicontinuous valued mapping F

### Popular passages

Page 163 - P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets and Systems 35 (1990) 241-249.