Fundamentals of Fuzzy Sets
Fundamentals of Fuzzy Sets covers the basic elements of fuzzy set theory. Its four-part organization provides easy referencing of recent as well as older results in the field. The first part discusses the historical emergence of fuzzy sets, and delves int
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AN INTRODUCTION TO FUZZY RELATIONS Sergei Ovchinnikov
Abstract. TIlis 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. TIlen, 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.
4.1 INTRODUCTION Crisp relations play an important role in mathematics and its applications to other sciences. In the area of pure mathematics, most mathematical structures can be, and very often are, described in terms of relations, see Bourbaki (1974), Summary of Results. The notions of equivalence and ordering relations are used practically in all fundamental mathematical constructions; the theory of ordered sets itself is a big and rapidly developing branch of pure mathematics. In general system theory, an abstract system is defined as a relation among possible inputs and outputs, see Mesarovic and Takahara (1989). Binary and general multidimensional crisp relations have also found natural applications in modeling various concepts inherent to 'soft' sciences like psychology, sociology, linguistic, etc. Many models in measurement and decision-making theories are based on the notion of a relation. Relations are also widely used in different branches of computer science. The idea to use numbers to describe relations is not a new one. In response to Poincare's critical thoughts regarding the "physical continuum" (poincare, 1912),
D. Dubois et al. (eds.), Fundamentals of Fuzzy Sets © Kluwer Academic Publishers 2000
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FUNDAMENTALS OF FUZZY SETS
Karl Menger suggested to use probabilistic indistinguishability relations as "a more realistic theoretical description of the equality of two elements", see Menger (1951). General probabilistic relations are important tools in measurement, decision-making, and choice theories, see Suppes et a1. (1989), section 16.4.2. Fuzzy set theory suggests a different approach to modeling relations. While a crisp relation represents just the presence or absence of interaction or association among elements of given sets, the values of the membership function of a fuzzy relation represent degrees of association. Thus crisp relations are simply a special case of fuzzy relations, like crisp sets are a restricted case of fuzzy sets. Lotfi Zadeh introduced the notion of a fuzzy binary relation in his first paper on fuzzy sets in Zadeh (1965), and defined such important concepts as similarity relations and fuzzy orderings in Zadeh (1971). The concept of a fuzzy relation is one of the most fundamental notions in fuzzy set theory and is a major mathematical tool in numerous application areas. In this chapter we introduce basic definitions and properties of fuzzy relations and describe main classes of fuzzy binary relations. The reader sh
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