Some basic theory of interval-valued fuzzy sets

Let S be a set. A fuzzy subset of S is a mapping A from S into [0,1]. The value A(s) for a particular s is typically associated with a degree of belief of some expert. There is an extensive theory of fuzzy sets, but there are situations when assigning an exact number to an expert\´s opinion is too restrictive, and the assignment of an interval of values is more realistic. This paper is concerned with the basics of a theory for such "interval-valued" fuzzy sets, namely mappings of a set S into the set of intervals of [0,1]