The book is designed as a textbook for the undergraduate and postgraduate students of Science and Engineering of various Indian universities on the basis of the University Grants Commission curriculum. Particularly, covers the entire ambit of the course Fuzzy Mathematics for Semester VI B.Tech. Program of Amity University. In this book, fuzzy concepts have been introduced as generalization and extension of crisp sets. Also emphasis is given not only to the presentation of the fundamental and theoretical concepts of fuzzy set theory in an intelligible and easy-to-understand manner but also details the theoretical advances and applications of fuzzy sets in the real world phenomena. This book provides students with a self-contained introduction that requires no preliminary knowledge of fuzzy mathematics. The book covers the following in 9 chapters:
Chapter 1 provides fundamentals of basic crisp set theory and the mapping of crisp set to a function. Chapter 2 deals with the introduction, basic definitions, standard operations and properties of fuzzy sets. It then proceeds to link the crisp set with fuzzy set theory and shows how and where these two theories concur as well as differ. Chapter 3 deals with the alpha cut set and the extent to which an element is a member of a fuzzy set, followed by the features and types of membership functions. Chapter 4 shows how fuzzy sets can be represented by family of crisp sets and also provides the extension principle on multiple fuzzy sets. Fuzzy complement function along with the Yager and Sugeno class of complement function is given in Chapter 5. Chapter 6 covers the axioms, types and characterization theorems of t-norms and t-conorms. Chapter 7 deals with the various aggregation operations on fuzzy sets. Chapter 8 provides basic aspects of fuzzy numbers and linguistic variables and also covers the arithmetic operations between fuzzy numbers with interval analysis method and with extension principle method. Finally, fuzzy equations are introduced and examined in chapter 9.
Each topic has been thoroughly covered in scope, content and also from the examination point of view. For each topic, several worked out examples, simple as well as typical, carefully selected to cover all aspects of the topic, so that the reader may gain confidence in the techniques of solving problems of each chapter.
- Crisp Sets
- Fuzzy Sets
- α -Cut Set and Membership Function
- Decomposition of a Fuzzy Set and Extension Principle
- Fuzzy Complement
- T-Norms and T-Conorms
- Aggregation Operations
- Arithmetic Operations on Fuzzy Numbers
- Fuzzy Equations
Tutorial Sheet 1
Tutorial Sheet 2
Tutorial Sheet 3
Tutorial Sheet 4