**Description****:**

This book presents the
rudiments of fuzzy set theory and fuzzy logic and related topics and their
applications in a simple and easy-to-understand manner. Written with a general
type of reader in mind, the book avoids the extremes of abstract mathematical proofs
as well as specialized technical details of different areas of application. The
book is intended to motivate the readers in getting a clear grasp of the basic
concepts of the subject and its possible applications. A special feature of the
book is that the exercises at the end of each chapter are preceded by a set of
short-answer review-type questions. These are intended to help the reader in
developing the confidence that he/she has correctly grasped the concepts
introduced in the chapter. The book can be used as a text for the study of the
topics of fuzzy set theory, fuzzy logic and their possible applications at the
undergraduate, graduate and postgraduate students of mathematics, engineering
and other disciplines of science, arts and medicine.

**In this book:**

• Introduction

• Fuzzy Sets

• Operations on Fuzzy Sets

• Fuzzy Numbers

• Fuzzy Relations and Fuzzy
Graphs

• Fuzzy Functions and Calculus of
Fuzzy Functions

• Measures of Vagueness and
Uncertainty

• Evidence Theory, Possibility
Theory and Probability Theory

• Linguistic Variables, Fuzzy
Rules and Fuzzy Propositions

• Fuzzy Logic and Approximate
Reasoning

• Decision Making in a Fuzzy
Environment

• Construction of Fuzzy Sets and
Defuzzification

• Fuzzy Rules Based Models and
Fuzzy Systems

• Engineering and Other
Applications

• Intutionistic Fuzzy Sets, Rough
Sets, Vague Sets and Soft Sets

**Contents****:**

*Preface to the Second Edition*

*Preface to the First Edition*

*Acknowledgements*

**Chapter 1. Introduction **•
Limitations of the Conventional Tools of Analysis • Certainty Versus
Uncertainty • Nature and Role of Uncertainty • Fuzzy Set Theory and Related
Concepts • About This Book

**Chapter 2. Fuzzy Sets **•
Classical Sets: An Overview • Extensional Definition • Intensional Definition •
Characteristic Functional Definition • Fuzzy Sets: Why do we need them • Fuzzy
Sets • Other Representations of Fuzzy Sets • There is nothing fuzzy about Fuzzy
Sets • Properties of Fuzzy Sets • Cardinality and relative cardinality • Height
• Support of a fuzzy set • a-cuts • Compliment of a Fuzzy Set • Equal Fuzzy
Sets • Universal Set, Empty Set and Crossover Point • Failing of the Law of
Excluded Middle and the Law of Contradiction • Subset of a Fuzzy Set • Convex
Fuzzy Sets • Geometric Interpretation of Fuzzy Sets • Features of Membership
Function • Some Well Known Membership Functions • Triangular Membership
Function • Trapezoidal Membership Function • Bell-shaped, Gaussian and
Sigmoidal membership Functions • Some other commonly used Membership Functions •
Short Answer Review Questions • Exercises

**Chapter 3. Operations on Fuzzy
Sets **• Introduction • Operations on Crisp Sets • Operations on FuzzySets •
Union and Intersection of Fuzzy Sets • Algebraic Sum, Bounded Sum and Bounded
Difference • Algebraic Product • Power of a Fuzzy Set • Some other Operations
on Fuzzy Sets • Convex Combination of Fuzzy Sets • Operations on Fuzzy Sets
Defined on Different Universes of Discourse • Cartesian Product • Extension
Principle • t-Norm and t-Conorm Operations • Certain t-norm and t-conorm
Operations • Pairs of t-norm and t-conorm Operations • Some well-known pairs of
t-norm and t-conorm Operations • Generation of t-norm and t-Conorm operations •
Short Answer Review Questions • Exercises

**Chapter 4. Fuzzy Numbers **•
Introduction • Fuzzy Numbers • Positive and Negative Fuzzy Numbers • Fuzzy
Numbers of *LR *Type • Triangular and Trapezoidal Fuzzy Numbers •
Arithmetic Operations on Fuzzy Numbers • Operations on Fuzzy Numbers on the
Basis of Extension Principle • Addition, Subtraction, Multiplication and
Division Operations • Operations on Fuzzy Numbers of LR type • Arithmetic
Operations on Fuzzy Numbers Based on a-Cuts • Ordering of Fuzzy Numbers • Fuzzy
Equations • Solution of Fuzzy Equations of Type *Ã *? *X? = B? *and Ã ? *X? *= *B? *• Solution of Fuzzy
Equations of the Type *Ã *? *X? = B?, Ã X? = B? *• Conditions for
Existence of Solutions • Short Answer Review Questions • Exercises

**Chapter 5. Fuzzy Relations and
Fuzzy Graphs **• Introduction • Fuzzy Relations on Crisp Sets • Fuzzy Relations
as Matrices • Fuzzy Relations on Fuzzy Sets • Domain and Range of a Fuzzy
Relation • Union and Intersection of Fuzzy Relations • Projections and
Cylindrical Extensions • Projection • Cylindrical Extension • Composition of
Fuzzy Relations • The max-min, max-product and max-average compositions •
Properties of max-min Composition • Relational Join • Symmetric and Anti
Symmetric Relations • Transitivity • Transitive Closure of a Fuzzy Relation *R*^{~}(X, X) •
Fuzzy Graph • Special Fuzzy Relations • Similarity Relations • Fuzzy Ordering
Relations • Fuzzy Compatibility Relations • Fuzzy Morphisms • Fuzzy Relation
Equations • Short Answer Review Questions • Exercises

**Chapter 6. Fuzzy Functions and
Calculus of Fuzzy Functions **• Introduction • Fuzzy Functions •
Correspondence between a Fuzzy Function and a Fuzzy Relation • Inverse of a
Fuzzy Function • Extrema of Fuzzy Functions • Integration of Fuzzy Functions •
Integration of a Fuzzy Function Over a Crisp Interval • Properties of Integrals
of Fuzzy Functions • Integration of a Crisp Real Valued Function Over a Fuzzy
Interval • Properties of Integrals of Crisp Functions Over Fuzzy Intervals •
Differentiation of a Fuzzy Function • Properties of Derivatives of Crisp
Functions at Fuzzy Points • Short Answer Review Questions • Exercises

**Chapter 7. Measures of
Vagueness and Uncertainty **• Introduction • Fuzzy Measures •
Difference between a Fuzzy Measure and a Measure of Fuzziness • Uncertainty
Based Information • Non - Specificity of Crisp Sets • Non-specificity of Fuzzy
Sets • Fuzziness of Fuzzy Sets • Fuzziness as Lack of Distinction Between a Set
and Its Complement • Non-specificity Versus Fuzziness • Short Answer Review
Questions • Exercises

**Chapter 8. Evidence Theory,
Possibility Theory and Probability Theory **• Introduction • Evidence
Theory • Possibility Theory • Possibility Distributions • Possibility Measure •
*U*-Uncertainty of Possibility Distributions • Joint Possibility
Distributions • Relationship Between Possibility Distributions and Probability
Distributions • Relationship Between Possibility and Necessity • Fuzzy Sets and
Possibility Theory • Possibility Theory Versus Probability Theory •
Non-interactiveness Versus Independence • Measures of Non-specificity in
Probability Theory • Probability of a Fuzzy Event • Conditional Probability of
Fuzzy Events • Independent Fuzzy Events • Bayes’ Theorem for Fuzzy Events •
Fuzzy Probability • Probabilistic Interpretation of Fuzzy Sets • Possibility
Measures and Probability Measures as Fuzzy Measures • Principle of Uncertainty
Invariance • Short Answer Review Questions • Exercises

**Chapter 9. Linguistic
Variables, Fuzzy Rules and Fuzzy Propositions **•
Introduction • Linguistic Variables • Linguistic Variables ‘Truth’ and
‘Probable’ • Hedges • Structured Linguistic Variables • Boolean Linguistic Variables
• Quantifiers • Fuzzy Rules • Fuzzy Mapping Rule • Fuzzy Implication Rules •
Fuzzy Propositions • Unconditional and Unqualified Propositions • Unconditional
and Qualified Propositions • Conditional and Unqualified Propositions •
Conditional and Qualified Propositions • Short Answer Review Questions •
Exercises

**Chapter 10. Fuzzy Logic and
Approximate Reasoning **• Introduction • Classical Logic: An Overview • Truth Tables of
Classical Logic • Modus Ponens and Modus Tolens • Propositional Logic • First
Order Predicate Calculus • Multivalued Logic • Fuzzy Logic • Fuzzy Truth Tables
• Baldwin’s Version of Fuzzy Logic • Inference from Conditional Fuzzy
Propositions • Inference from Conditional and Quantified Propositions •
Inference from Fuzzy Quantified Propositions • Fuzzy Implications • Fuzzy
Implication Rules • Criteria of Fuzzy Implications • Families of Fuzzy
Implications • Approximate Reasoning • *Short Answer Review *Questions •
Exercises

**Chapter 11. Decision Making in
a Fuzzy Environment **• Introduction • Individual Decision Making • Trade-off Between
Goals and Constraints • Multiperson Decision Making • Multicriteria Decision
Making • Multiperson-Multicriteria Decision Making • Fuzzy Ranking of
Alternatives • Optimization in a Fuzzy Environment • Fuzzy Linear Programming •
Multistage Decision Making • Short Answer Review Questions • Exercises

**Chapter 12. Construction of
Fuzzy Sets and Defuzzification **• Introduction • Necessity for
Fuzzification • Construction of Fuzzy Sets • Methods for Constructing
Membership Functions • Direct Methods with Single Expert • Direct Methods with
Multiple Experts • Indirect Methods with Single Expert • Indirect Methods with
Multiple Experts • Construction from Sample Data • Fusion of Local Models •
Defuzzification • Mean of Maximum (MOM) Method • Centre-of-Area (COA) Method •
The Height Method • Centre of Maxima Method • Parameterized Family of
Defuzzification Methods • *Short Answer Review *Questions • Exercises

**Chapter 13. Fuzzy Rules Based
Models and Fuzzy Systems **• Introduction • Fuzzy Rules Based
Inference • Fuzzy Rules Based Models • Fuzzy Partition • Mapping a Fuzzy
Subspace to Local Models • Different Types of Fuzzy Rule-based Models • The
Mamdani Model • The TSK Model • Standard Additive Model • Fuzzy Systems • Fuzzy
Expert Systems • Fuzzy Controllers • A Fuzzy Controller • Fuzzy Automata •
Short Answer Review Questions • Exercises

**Chapter 14. Engineering and
Other Applications **• Introduction • Engineering Applications • Civil Engineering •
Mechanical Engineering • Industrial Engineering • Electrical Engineering •
Computer Engineering • Reliability Theory • Pattern Recognition, Cluster
Analysis and Image Processing • Robotics • Other Engineering Applications •
Applications in Other Disciplines • Medicine • Economics • Fuzzy Regression •
Genetic Algorithms • Neural Networks • Soft Computing • Interpersonal
Communication • Natural Sciences • Psychology • Languages • Miscellaneous •
Exercises

**Chapter 15. Intuitionistic
Fuzzy Sets, Rough Sets, Vague Sets and Soft Sets **•
Introduction • Intuitionistic Fuzzy Sets • Special Classes of Intuitionistic
Fuzzy Sets • (a, ß) cut of an IFS • Construction of an Intuitionistic Fuzzy Set
from a Fuzzy Set • Intuitionistic Fuzzy Numbers (IFN) • Arithmetic Operations
onIFN’s • Vague Sets • Difference between IFS and VS • Algebraic
Operations on Vague Sets • Soft Sets • Operation on Soft Sets • Fuzzy Soft Sets
• Vague Soft Sets • Rough Sets • Vagueness and Uncertainty in Rough Sets •
Uncertainty of Rough Sets • Fuzzy Rough Sets and Rough Fuzzy Sets • *Short
Answer Review *Questions • Exercises

*Bibliography*

Index

**About ****the****Author****:**

**Prof. Chander Mohan **was born
on 8 October 1939 in Jhelum, British India, now on Pakistan. He completed his
B.A. (Honours) in Mathematics in 1957 and M.A. in Mathematics in 1960 from
Panjab University, Chandigarh, India. He obtained his Ph.D. in Mathematics from
the University of Roorkee, India in 1967. He taught at Multanimal Modi College,
Modinagar, India, as Lecturer in Mathematics during 1961–64. Thereafter he
taught at the Department of Mathematics, University of Roorkee, as Lecturer
(1964–70); as Reader (1970–85); and as Professor (1985–2000). He was also Head
of the Department during 1994–99. He has also served as Professor and Dean,
Amity School of Computer Science, Noida, Uttar Pradesh, India (2000–02); as
Professor and Head, Mathematics Department, I.I.L.M. College of Engineering and
Technology, Greater Noida, Uttar Pradesh, India (2002–04). Currently, he is
Professor Emeritus, Department of Computer Science, Ambala College of
Engineering and Applied Research, Ambala, Haryana, where he has been teaching
since 2004. He has over 90 research publications in national and international
research journals and three books on technical subjects to his credit. He is
also the recipient of the Khosla Research Award in 1990 and Millennium Award of
Honour of the American Biographical Institute. He is listed in the Who’s Who
publications. He is a member of the International Astronomical Union,
Operations Research Society of India (Founder Member), Indian Astronomical
Society (Founder Member), Indian Society for Industrial and Applied Mathematics
(Executive Committee Member), and honorary chair, Indian Society of Soft
Computing.

**Target
Audience:**

The book can be used as a text
for the study of the topics of fuzzy set theory, fuzzy logic and related
concepts and their applications both at the undergraduate and postgraduate
level in mathematics and other disciplines of science, arts, engineering and
medicine.