A First Course in Fuzzy Logic

Hung T. Nguyen, Carol Walker, Elbert A. Walker

• Publisher: Chapman and Hall/CRC
• Year: 2018
• ISBN:
  9780429012600
• Edition: 4
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

A First Course in Fuzzy Logic, Fourth Edition is an expanded version of the successful third edition. It provides a comprehensive introduction to the theory and applications of fuzzy logic.

This popular text offers a firm mathematical basis for the calculus of fuzzy concepts necessary for designing intelligent systems and a solid background for readers to pursue further studies and real-world applications.

New in the Fourth Edition:

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• Features new results on fuzzy sets of type-2

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• Provides more information on copulas for modeling dependence structures

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• Includes quantum probability for uncertainty modeling in social sciences, especially in economics

With its comprehensive updates, this new edition presents all the background necessary for students, instructors and professionals to begin using fuzzy logic in its many—applications in computer science, mathematics, statistics, and engineering.

About the Authors:

Hung T. Nguyen is a Professor Emeritus at the Department of Mathematical Sciences, New Mexico State University. He is also an Adjunct Professor of Economics at Chiang Mai University, Thailand.

Carol L. Walker is also a Professor Emeritus at the Department of Mathematical Sciences, New Mexico State University.

Elbert A. Walker is a Professor Emeritus, Department of Mathematical Sciences, New Mexico State University.

Chapter 1

THE CONCEPT OF FUZZINESS

In this opening chapter, we will discuss the intrinsic notion of fuzziness in natural language. Following Lotfi Zadeh, fuzzy concepts will be modeled as fuzzy sets, which are generalizations of ordinary (crisp) sets.

1.1 Examples

In using our everyday natural language to impart knowledge and information, there is a great deal of imprecision and vagueness, or fuzziness. Such statements as “John is tall” and “Fred is young” are simple examples. Our main concern is representing, manipulating, and drawing inferences from such imprecise statements.

We begin with some examples.

Example 1.1.1 The description of a human characteristic such as healthy;

Example 1.1.2 The classification of patients as depressed;

Example 1.1.3 The classification of certain objects as large;

Example 1.1.4 The classification of people by age such as old;

Example 1.1.5 A rule for driving such as “if an obstacle is close, then brake immediately”.

In the examples above, terms such as depressed and old are fuzzy in the sense that they cannot be sharply defined. However, as humans, we do make sense out of this kind of information, and use it in decision making. These “fuzzy notions” are in sharp contrast to such terms as married, over 39 years old, or under 6 feet tall. In ordinary mathematics, we are used to dealing with collections of objects, say certain subsets of a given set such as the subset of even integers in the set of all integers. But when we speak of the subset of depressed people in a given set of people, it may be impossible to decide whether a person is in that subset or not. Forcing a yes-or-no answer is possible and is usually done, but there may be information lost in doing so because no account is taken of the degree of depression. Although this situation has existed from time immemorial, the dominant context in which science is applied is that in which statements are precise (say either true or false)—no imprecision is present. But in this time of rapidly advancing technology, the dream of producing machines that mimic human reasoning, which is usually based on uncertain and imprecise information, has captured the attention of many scientists. The theory and application of fuzzy concepts are central in this endeavor but remain to a large extent in the domain of engineering and applied sciences.

With the success of automatic control and of expert systems, we are now witnessing an endorsement of fuzzy concepts in technology. The mathematical elements that form the basis of fuzzy concepts have existed for a long time, but the emergence of applications has provided a motivation for a new focus for the underlying mathematics. Until the emergence of fuzzy set theory as an important tool in practical applications, there was no compelling reason to study its mathematics. But because of the practical significance of these developments, it has become important to study th...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 The Concept of Fuzziness
• 2 Some Algebra of Fuzzy Sets
• 3 Fuzzy Quantities
• 4 Logical Aspects of Fuzzy Sets
• 5 Basic Connectives
• 6 Additional Topics on Connectives
• 7 Fuzzy Relations
• 8 Universal Approximation
• 9 Possibility Theory
• 10 Partial Knowledge
• 11 Fuzzy Measures
• 12 The Choquet Integral
• 13 Fuzzy Modeling and Control
• Bibliography
• Answers to Selected Exercises
• Index