Preface

In mathematics, we investigate the properties of abstract objects (e.g. numbers, geometric shapes, spaces) and the possible relationships between these objects. One basic characteristic of mathematics is that all these properties and relationships are absolute. Thus, properties and relationships are either true or false. Nothing else is meaningful. In other words, in mathematics there is no room for vagueness, for randomness, and for extremely small quantities. By introducing one of these qualities into mathematics, one can create alternative mathematics [1]. But how do we introduce vagueness into mathematics? One very simple way to achieve this is to allow notions like “small,” “large,” and “few.” However, another way is to modify the most basic object of mathematics, that is, to modify sets. In this respect, fuzzy mathematics is a form of alternative mathematics since it is based on a generalization of set membership. Simply put, in fuzzy mathematics, an element may belong to a degree to a set, while in ordinary mathematics, it either belongs or does not belong to a set. This simple idea has been applied to most fields of mathematics and so we can talk about fuzzy mathematics.

Even today, many researchers and thinkers consider fuzzy mathematics as a tool that can be used instead of probability theory to reason about or to work with a specific system. This text is based on the idea that vagueness is a basic notion and thus tries to present fuzzy mathematics as a form of alternative mathematics and not as an alternative to probability theory. In addition, this text is an introduction to fuzzy mathematics. This simply means that we have tried to cover as many fields of fuzzy mathematics as possible. Thus, this text is a compendium but not a handbook of fuzzy mathematics.

Chapter 1 of this book explains what vagueness means from a philosophical point of view. Also, it demonstrates the connection between vagueness and fuzzy mathematics.

In Chapter 2, we introduce the notion of fuzzy set as well as the basic set operations. In addition, we introduce a number of variants or extensions of fuzzy sets. The chapter concludes with a section marked with a star. There are a few such sections in the book and these are optional readings as they deal with quite advanced ideas.

Fuzzy number are special kinds of fuzzy sets that, in a way, have been introduced to generalize the notion of a number. In Chapter 3, we present various forms of fuzzy numbers and the basic arithmetic operations between them. We introduce linguistic variables, that is, terms such as “small,” “heavy,” “tall.” Also, we present fuzzy equations (e.g. simple equations like images, where known and unknown quantities are fuzzy numbers) and how one can solve them. The chapter concludes with applications of fuzzy numbers. In general, most chapters that follow have a final section that presents applications of the main material presented in the specific chapter.

Fuzzy relations are a very important subject that is presented in Chapter 4. We discuss fuzzy relations, the Cartesian product of fuzzy sets and related notions, and fuzzy orders. Since graphs can be described by relations, we also discuss fuzzy graphs. Also, since precategories are also described by graphs, we discuss fuzzy categories.

Chapter 5 is devoted to possibility theory, that is, generally speaking, the fuzzy “version” of a probability theory. Also, we compare probability and possibility theories in order to see their differences.

Chapter 6 discusses fuzzy statistics. In this chapter, we discuss fuzzy random variables (in a way as something that corresponds to vague randomness …) and all related notions such as fuzzy regression and fuzzy point estimation.

In Chapter 7, we discuss many‐valued and fuzzy logics. We do not just present truth values and the basic logical operations, but we present complete logical systems. In addition, we discuss approximate reasoning and try to see what is a logic of vagueness.

Although computability theory is a very basic part of logic, we discuss in a separate chapter fuzzy computation (Chapter 8). In particular, we discuss fuzzy automata, fuzzy Turing machines, and other fuzzy models of computation.

In Chapter 9, we give a taste of fuzzy abstract algebra theory. We present fuzzy groups, fuzzy rings, fuzzy vector spaces, fuzzy normed spaces, and fuzzy Lie algebras.

Chapter 10 introduces the basic notions and ideas of fuzzy metric spaces and fuzzy topology. In addition, we briefly discuss fuzzy Banach spaces and fuzzy Hilbert spaces.

Fuzzy geometry is introduced in Chapter 11. We discuss the notion of fuzzy points and the distance between them, fuzzy lines, fuzzy circles, and fuzzy polygons.

Chapter 12 introduces the reader to fuzzy calculus. In particular, we discuss fuzzy functions, integrals and derivatives of fuzzy functions, and fuzzy limits of sequences and function. Furthermore, fuzzy (ordinary and partial) differential equations are presented.

The book includes two appendices: the first briefly presents fuzzy approximation and the second gives a taste of fuzzy chaos and fuzzy fractals.

Each chapter starts with a section that describes the nonfuzzy concepts whose fuzzy counterparts are presented in the rest of the chapter. We felt this was necessary since we present many and quite diverse topics and we cannot expect everyone to be familiar with all these notions and ideas. Also, most chapters have some exercises at the end. Readers are invited to work on them if they want to deepen their understanding of the ideas presented in the corresponding chapter. However, the chapter on fuzzy computation does not include exercises since the subject is not mature enough.

Apostolos Syropoulos
Xanthi, Greece, November 2019

Theophanes Grammenos
Volos, Greece, November 2019

Reference

  1. 1   Van Bendegen, J.P. (2005). Can there be an alternative mathematics, really? In: Activity and Sign: Grounding Mathematics Education (ed. M.H. Hoffmann, J. Lenhard, and F. Seeger), 349–359. Boston, MA: Springer US.
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