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Mathematics of Fuzzy Sets Logic, Topology, and Measure Theory by Ulrich Höhle

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Published by Springer US in Boston, MA .
Written in English

Subjects:

  • Mathematics,
  • Mathematical optimization,
  • Symbolic and mathematical Logic,
  • Operations research

Book details:

About the Edition

Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory is a major attempt to provide much-needed coherence for the mathematics of fuzzy sets. Much of this book is new material required to standardize this mathematics, making this volume a reference tool with broad appeal as well as a platform for future research. Fourteen chapters are organized into three parts: mathematical logic and foundations (Chapters 1-2), general topology (Chapters 3-10), and measure and probability theory (Chapters 11-14). Chapter 1 deals with non-classical logics and their syntactic and semantic foundations. Chapter 2 details the lattice-theoretic foundations of image and preimage powerset operators. Chapters 3 and 4 lay down the axiomatic and categorical foundations of general topology using lattice-valued mappings as a fundamental tool. Chapter 3 focuses on the fixed-basis case, including a convergence theory demonstrating the utility of the underlying axioms. Chapter 4 focuses on the more general variable-basis case, providing a categorical unification of locales, fixed-basis topological spaces, and variable-basis compactifications. Chapter 5 relates lattice-valued topologies to probabilistic topological spaces and fuzzy neighborhood spaces. Chapter 6 investigates the important role of separation axioms in lattice-valued topology from the perspective of space embedding and mapping extension problems, while Chapter 7 examines separation axioms from the perspective of Stone-Cech-compactification and Stone-representation theorems. Chapters 8 and 9 introduce the most important concepts and properties of uniformities, including the covering and entourage approaches and the basic theory of precompact or complete [0,1]-valued uniform spaces. Chapter 10 sets out the algebraic, topological, and uniform structures of the fundamentally important fuzzy real line and fuzzy unit interval. Chapter 11 lays the foundations of generalized measure theory and representation by Markov kernels. Chapter 12 develops the important theory of conditioning operators with applications to measure-free conditioning. Chapter 13 presents elements of pseudo-analysis with applications to the Hamilton & endash;Jacobi equation and optimization problems. Chapter 14 surveys briefly the fundamentals of fuzzy random variables which are [0,1]-valued interpretations of random sets.

Edition Notes

Statementedited by Ulrich Höhle, Stephen Ernest Rodabaugh
SeriesThe Handbooks of Fuzzy Sets Series -- 3, Handbooks of fuzzy sets series -- 3.
ContributionsRodabaugh, Stephen Ernest
Classifications
LC ClassificationsQA8.9-10.3
The Physical Object
Format[electronic resource] :
Pagination1 online resource (xii, 716 pages).
Number of Pages716
ID Numbers
Open LibraryOL27073386M
ISBN 101461373107, 1461550793
ISBN 109781461373100, 9781461550792
OCLC/WorldCa851819273

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Mathematics of Fuzzy Sets and Fuzzy Logic. Abstract. This book presents a mathematically-based introduction into the fascinating topic of Fuzzy Sets and Fuzzy Logic and might be used as textbook at both undergraduate and graduate levels and also as reference guide for mathematician, scientists or engineers who would like to get an. Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory is a major attempt to provide much-needed coherence for the mathematics of fuzzy sets. Much of this book is new material required to standardize this mathematics, making this volume a reference tool with broad appeal as well as a platform for future research. In the final section of the chapter, nonstandard fuzzy sets are briefly discussed. As the authors explain, they decided not to discuss various forms of nonstandard fuzzy sets in detail to keep the size of the book within reasonable limits. Nevertheless, some corresponding concepts are covered with interesting historical background information. The Fuzzy Set Theory section of Mathematics aims at disseminating and communicating fuzzy set theory driven scientific knowledge and impactful discoveries to academia, industry, and the public worldwide. The concept of a fuzzy set, on which fuzzy logic (FL) has been built, has been proven to play an important role in (1) modeling and representing imprecise and uncertain linguistic human.

  The second to last chapter examines the application of fuzzy and intuitionistic fuzzy mathematics in image enhancement, segmentation, and retrieval. Finally, the book concludes with coverage the extension of fuzzy sets. This book: Covers both fuzzy and intuitionistic fuzzy sets and includes examples and practical applications. The target of the present Special Issue of the MDPI journal Mathematics is to provide the experts in the field (academics, researchers, practitioners, etc.) the opportunity to present recent theoretical advances on fuzzy sets and fuzzy logic and of their extension/generalization (e.g. intuitionistic fuzzy logic, neutrosophic sets, etc.) and. "This book presents a mathematically-based introduction into the fascinating topic of Fuzzy Sets and Fuzzy Logic and might be used as textbook at both undergraduate and graduate levels and also as reference guide for mathematician, scientists or engineers who . Set Theory by Anush Tserunyan. This note is an introduction to the Zermelo–Fraenkel set theory with Choice (ZFC). Topics covered includes: The axioms of set theory, Ordinal and cardinal arithmetic, The axiom of foundation, Relativisation, absoluteness, and reflection, Ordinal definable sets and inner models of set theory, The constructible universe L Cohen's method of forcing, Independence.

Fuzzy Mathematics. Fuzzy logic is an extension or a superset of the Boolean logic – aimed at maintaining the concept of the “partial truth,” i.e. expression values ranging from “completely truthful” to “completely untruthful” (from 0 to 1).   In the mid'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's, I had suggested similar ideas about continuous-truth-valued propositional calculus (inffor "and", sup for "or") to my. Other articles where Fuzzy set is discussed: fuzzy logic: Fuzzy sets: Most concepts used in everyday language, such as “high temperature,” “round face,” or “aquatic animal,” are not clearly defined. In Lotfi Zadeh, an engineering professor at the University of California at Berkeley, proposed a mathematical definition of those classes that lack precisely. Mathematics of Fuzzy Sets and Fuzzy Logic Barnabas Bede This book presents a mathematically-based introduction into the fascinating topic of Fuzzy Sets and Fuzzy Logic and might be used as textbook at both undergraduate and graduate levels and also as reference guide for mathematician, scientists or engineers who would like to get an insight.