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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].