Synopses & Reviews
34105-7
From the beginning of modern science until the end of the nineteenth century, uncertainty was generally viewed as undesirable in science, but with the emergence of statistical mechanics in the twentieth century, the unmanageable complexity of mechanical processes on the molecular level led to the adoption of statistical principles and principles of probability theory. However, in spite of its success, probability theory alone is not capable of capturing uncertainty in all of its manifestations, particularly when it arises from the vagueness of natural language. Thus, a new theory arose which treats this aspect of uncertainty: fuzzy set theory.
Fuzzy Set Theory: Foundations and Applications serves as a simple introduction to basic elements of fuzzy set theory. The emphasis is on a conceptual rather than a theoretical presentation of the material. Fuzzy Set Theory also contains an overview of the corresponding elements of classical set theory-including basic ideas of classical relations-as well as an overview of classical logic. Because the inclusion of background material in these classical foundations provides a self-contained course of study, students from many different academic backgrounds will have access to this important new theory.
- Introduction to the Concept of Fuzziness
- Classical Logic
- Classical Set Theory
- Fuzzy Sets
- Classical Relations
- Fuzzy Relations
- Fuzzy Arithmetic
- Fuzzy Logic
- Applications: A Survey
“This user-friendly text is a valuable resource to introduce professionals from many disciplines to the broad applicability of fuzzy set theory and fuzzy logic to many areas of human affairs. Electrical, mechanical, and software engineers, chemists, and managers will find this a useful tool for learning about basic principles and ideas of the increasingly important fuzzy set theory which later could progress to its application in their specific fields. This book would be appropriate as a textbook for a general course in undergraduate liberal arts and sciences programs as a meaningful enrichment of a typical course on the basics of classical set theory and classical logic, and as a reference source for introductory short courses for working professionals.” Prof. Marian Stachowicz, Jack Rowe Chair, Dept. of Computer Eng., Univ. of Minnesota
“The text provides a very carefully crafted introduction to the basics of fuzzy sets.” , Prof. Les Sztandera, Dept. of Computer Science, Philadelphia College of Textiles and Sciences
0-13-341058-7
Synopsis
This book is designed to help anyone understand the basics of fuzzy sets, whether or not they have a mathematical background. The book first presents a basic grounding in information theory, classical logic and set theories. Next, it introduces the basics of fuzzy sets, distinguishing them from traditional crisp sets, and introducing the concept of membership function. The distinctions between classical and fuzzy relations are introduced, as are representations of fuzzy relations; fuzzy equivalence relations; fuzzy partial orderings, and related topics. The book introduces fuzzy arithmetic and fuzzy numbers. It also presents a detailed introduction to fuzzy logic, multivalued logics, fuzzy propositions, quantifiers, linguistic hedges and approximate reasoning. Several basic and advanced applications for fuzzy set theory are presented as well. Any non-technical reader interested in fuzzy sets and fuzzy logic. Also ideal for introductory level-students, whether they are planning a technical or non-technical course of study.
Synopsis
This book is designed to help anyone understand the basics of fuzzy sets, whether or not they have a mathematical background.The book first presents a basic grounding in information theory, classical logic and set theories. Next, it introduces the basics of fuzzy sets, distinguishing them from traditional crisp sets, and introducing the concept of membership function. The distinctions between classical and fuzzy relations are introduced, as are representations of fuzzy relations; fuzzy equivalence relations; fuzzy partial orderings, and related topics. The book introduces fuzzy arithmetic and fuzzy numbers. It also presents a detailed introduction to fuzzy logic, multivalued logics, fuzzy propositions, quantifiers, linguistic hedges and approximate reasoning. Several basic and advanced applications for fuzzy set theory are presented as well.Any non-technical reader interested in fuzzy sets and fuzzy logic. Also ideal for introductory level-students, whether they are planning a technical or non-technical course of study.
Table of Contents
Preface.
Introduction.
Information, Uncertainty, and Complexity.
Measurement and Uncertainty. Language and Vagueness. The Emergence of Fuzzy Set Theory. Fuzzy Set Theory Versus Probability Theory.
Classical Logic.
Introduction. Propositional Logic. Predicate Logic. Classical Set Theory.
Basic Concepts and Notation.
Set Operations. Fundamental Properties. Characteristic Functions of Crisp Sets. Other Concepts.
Fuzzy Sets: Basic Concepts and Properties.
Restrictions of Classical Set Theory and Logic. Membership Functions. Representations of Membership Functions. Constructing Fuzzy Sets. Operations on Fuzzy Sets.
Fuzzy Sets: Further Properties.
a-Cuts of Fuzzy Sets. a-Cut Representation. Cutworthy Properties of Fuzzy Sets. Extension Principle. Measurement of Fuzziness.
Classical Relations.
Introduction. Representations. Equivalence Relations. Partial Orderings. Projections and Cylindric Extensions.
Fuzzy Relations.
Introduction. Representations. Operations on Binary Fuzzy Relations. Fuzzy Equivalence Relations and Compatibility Relations. Fuzzy Partial Orderings. Projections and Cylindric Extensions. Fuzzy Arithmetic. Fuzzy Numbers. Arithmetic Operations on Intervals. Arithmetic Operations on Fuzzy Numbers.
Fuzzy Logic.
Introduction. Multivalued Logics. Fuzzy Propositions. Fuzzy Quantifiers. Linguistic Hedges. Approximate Reasoning.
Applications: A Survey.
An Historical Overview.
Established Applications.
Prospective Applications.
Illustrative Examples.
References for Applications.