A Modern Introduction to Fuzzy Mathematics

This book aims to fill a gap in fuzzy mathematics literature by presenting recent developments and also discussing what has been done so far in the field. The book contains 12 chapters. Chapter 1 introduces the reader to the basic ideas behind fuzzy sets. The authors explain what vagueness is and how it is different from other related ideas such as imprecision. Chapter 2 introduces the various forms, as well as the basic operations between fuzzy sets. In addition, the authors discuss the extension principle and introduce fuzzy numbers and the basic arithmetic operations between them. Next, Chapter 3 covers fuzzy relations and discusses their properties, while Chapter 4 introduces the notions of fuzzy predicate, fuzzy logics, ordinary fuzzy logical connectives, as well as t-norms and t-conorms. Fuzzy proof theory and related ideas are also covered in this chapter. Generalized measure theory is discussed in Chapter 5, while Chapter 6 introduces a fuzzy version of a statistical theory. Fuzzy grammars, fuzzy languages, and fuzzy automata are covered in Chapter 7, while Chapter 8 examines fuzzy Turing machines. Next, Chapter 9 discusses fuzzy multisets, while Chapter 10 introduces fuzzy algebraic structures and related ideas. Chapter 11 covers fuzzy calculus. Finally, in Chapter 12 the authors examine fuzzy topology and the properties of fuzzy topological spaces, as well as fuzzy metric spaces.