Interval-Valued Fuzzy Relations

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.