Fundamentals of Fuzzy Sets

“There is a fairly wide gap between what might be regarded as ‘animate’ system theorists and ‘inanimate’ system theorists at the present time, and it is not at all certain that this gap will be narrowed, much less closed, in the near future. There are some who feel that this gap reflects the fundamental inadequacy of conventional mathematics -the mathematics of precisely- defined points, functions, sets, probability measures, etc.- for coping with the analysis of biological systems, and that to deal effectively with such systems, which are generally orders of magnitude more complex than man-made systems, we need a radically different kind of mathematics, the mathematics of fuzzy or cloudy quantities which are not describable in terms of probability distributions. Indeed, the need for such mathematics is becoming increasingly apparent even in the realm of inanimate systems, for in most practical cases the a priori data as well as the criteria by which the performance of a man-made system are judged are far from being precisely specified or having accurately-known probability distributions”.

This paper is an introduction to fuzzy set theory. It has several purposes. First, it tries to explain the emergence of fuzzy sets from an historical perspective. Looking back to the history of sciences, it seems that fuzzy sets were bound to appear at some point in the 20th century. Indeed, Zadeh’s works have cristalized and popularized a concern that has appeared in the first half of the century, mainly in philosophical circles. Another purpose of the paper is to scan the basic definitions in the field, that are required for a proper reading of the rest of the volume, as well as the other volumes of the Handbooks of Fuzzy Sets Series. This Chapter also contains a discussion on operational semantics of the generally too abstract notion of membership function. Lastly, a survey of variants of fuzzy sets and related matters is provided.

This chapter summarizes main ways to extend classical set-theoretic operations (complementation, intersection, union, set-difference) and related concepts (inclusion, quantifiers) for fuzzy sets. Since these extensions are mainly pointwisely defined, we review basic results on the underlying unary or binary operations on the unit interval such as negations, t-norms, t-conorms, implications, coimplications and equivalences. Some strongly related connectives (means, OWA, weighted, and prioritized operations) are also considered, emphasizing the essential differences between these and the formerly investigated operator classes. We also show other operations which have no counterpart in the classical theory but play some important role in fuzzy sets (like symmetric sums, weak t-norms and conorms, compensatory AND).

This chapter presents a review of various interpretations of the fuzzy membership function together with ways of obtaining a membership function. We emphasize that different interpretations of the membership function call for different elicitation methods. We try to make this distinction clear using techniques from measurement theory.

This chapter presents an introduction to the theory of fuzzy relations. First, proximity and similarity relations, their classes and fuzzy partitions are introduced and their main properties are investigated. Then, main classes of fuzzy orderings are defined and classified with respect to the duality relations. Transitivity properties of fuzzy orderings and the Ferrers property of induced fuzzy orderings are established. Finally, we present elementary versions of two representation theorems.

This Chapter presents an overview of the different aspects of the concept of fuzzy equivalence relation (FER) as the extension to the fuzzy framework of the classical idea of equivalence. In this setting, new concepts like generator, dimension and base arise naturally. On the other hand, these relations can be dualy related with some kind of generalized metrics that allows a metric-like study of their properties. This chapter starts introducing some general ideas extending, for any triangular continuous norm, the concept of similarity relation already presented in Chapter 4. Then, we explain different methods for its effective construction. The relationship between fuzzy equivalence relations and generalized metrics is also studied. Next, based on the representation theorem, the concepts of generator, dimension and base are introduced. The structure of the generators set is studied and some procedures for calculating bases are presented.

Fuzzy relational equations are without doubt the most important inverse problems arising from fuzzy set theory, and in particular from fuzzy relational calculus. Indeed, the calculus of fuzzy relations is a powerful one, with applications in fuzzy control and fuzzy systems modelling in general, approximate reasoning, relational databases, clustering, etc. In this chapter, fuzzy relational equations are approached from an order-theoretical point of view. It is shown how all inverse problems can be reduced to systems of polynomial lattice equations. The exposition is limited to the description of exact solutions, and analytical ways are presented for obtaining the complete solution set when working in a broad and interesting class of distributive lattices. Ample literature pointers to approximate solution methods and application areas are provided.

Possibility theory was coined by L.A. Zadeh in the late seventies as an approach to model flexible restrictions constructed from vague pieces of information, described by means of fuzzy sets. Possibility theory is also a basic non-classical theory of uncertainty, different from but related to probability theory. This chapter discusses the basic elements of the theory: possibility and necessity measures (as well as two other set functions associated with a possibility distribution), the minimal specificity principle which underlies the whole theory, the notions of possibilitic conditioning and possibilistic independence, the combination, and projection of joint possibility distributions, as well as the possibilistic counterparts to integration. The relations and differences between this approach and other uncertainty frameworks, and especially probability theory, are pointed out. The difference between probability theory and fuzzy set theory is thus tentatively clarified. Lastly, decision-theoretic justifications of possibility theory are given.

This chapter is an overview of current research regarding measures of uncertainty and information in five mathematical theories: classical set theory, fuzzy set theory, probability theory, possibility theory, and the Dempster-Shafer theory of evidence. Three types of uncertainty are involved: nonspecificity, entropy-like uncertainty, and fuzziness.

In this chapter we discuss two facets of uncertainty, namely, fuzziness and non-Specificity, that may be associated with fuzzy sets. After presenting a set of desirable axioms that a measure of fuzziness should satisfy various attempts to quantify fuzziness are discussed. In this regard, the need for and quantification of higher order measures of fuzziness and weighted fuzziness are also presented. Finally, we discuss a few attempts to quantify non-specificity.

This chapter is an overview of past and present works dealing with fuzzy intervals and their operations. A fuzzy interval is a fuzzy set in the real line whose level-cuts are intervals. Particular cases include usual real numbers and intervals. Usual operations on the real line canonically extend to operations between fuzzy quantities, thus extending the usual interval (or error) analysis to membership functions. What is obtained is a counterpart of random variable calculus, but where, contrary to the latter case, there is no compensation between variables. Many results pertaining to basic properties of fuzzy interval analysis are summed up in the chapter. Computational methods are presented, exact or approximate ones, based on parametric representations, or level-cut approximations. The generalized fuzzy variable calculus involving interactive variables is also discussed with emphasis on triangular-norm based fuzzy additions. Dual ‘optimistic’ operations on fuzzy intervals, i.e., with maximal error compensation are also presented; its interest lies in providing tools for solving fuzzy interval equations. This chapter also contains a reasoned survey of methods for comparing and ranking fuzzy intervals. The chapter includes some historical background, as well as pointers to applications in mathematics and engineering.

This chapter gives an overview of distances between fuzzy numbers and the topology that these metrics induce. The metric structure allows the development of fuzzy analysis and various applications to interpolation, approximation and differential equations.