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## Über dieses Buch

This book offers an introduction to fuzzy sets theory and their operations, with a special focus on aggregation and negation functions. Particular attention is given to interval-valued fuzzy sets and Atanassov’s intuitionistic fuzzy sets and their use in uncertainty models involving imperfect or unknown information. The theory and application of interval-values fuzzy sets to various decision making problems represent the central core of this book, which describes in detail aggregation operators and their use with imprecise data represented as intervals. Interval-valued fuzzy relations, compatibility measures of interval and the transitivity property are thoroughly covered. With its good balance between theoretical considerations and applications of originally developed algorithms to real-world problem, the book offers a timely, inspiring guide to mathematicians and engineers developing new decision making models or implementing/applying existing ones to a wide range of applications involving imprecise or incomplete data.

## Inhaltsverzeichnis

### Chapter 1. Introduction to Fuzzy Sets

Abstract
In this chapter we will present basic notions and definitions constituting the basis of fuzzy set theory. Moreover, particular attention has been paid to the some extensions of fuzzy set theory, which involves imperfect or unknown situations as the uncertainty models. The definition and basic properties of these types of fuzzy sets are presented.
Barbara Pȩkala

### Chapter 2. Interval-Valued Fuzzy Relations

Abstract
In traditional fuzzy logic, to represent, e.g., the experts degree of certainty in different statements, numbers from the interval [0, 1] are used. It is often difficult for an expert to exactly quantify his or her certainty; therefore, instead of a real number, it is more adequate to represent this degree of certainty by an interval. In the first case, we get an interval-valued fuzzy set. In the second case, we get a second-order fuzzy set. Interval-valued fuzzy sets have been actively used in real-life applications. In this chapter the basic operations and properties of interval-valued fuzzy relations are considered. These properties of interval-valued fuzzy relations involve the notion of interval-valued fuzzy aggregations. The important issue will be examinations of properties of the generalized composition of interval-valued fuzzy relations.
Barbara Pȩkala

### Chapter 3. Applications

Abstract
Fuzzy sets and fuzzy logic are powerful mathematical tools for modeling and controlling uncertain systems in industry, humanity, and nature; they are facilitators for approximate reasoning in decision making in the absence of complete and precise information. Their role is significant when applied to complex phenomena not easily described by traditional mathematics. But due their limitations in the representation of reality that contains uncertainty or imprecision, we will focus on applying their extensions, especially the interval-valued fuzzy sets theory.
Barbara Pȩkala

### Chapter 4. Summary and Open Problems

Abstract
This chapter contains a short summary of the book’s contents, related open problems and proposals for future research.
Barbara Pȩkala

### Backmatter

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