Vague set

In mathematics, vague sets are an extension of fuzzy sets. 
In a fuzzy set, each object is assigned a single value in the interval [0,1] reflecting its "grade of membership". This single value does not allow a separation of evidence for membership and evidence against membership. 
Gau et al. proposed the notion of vague sets, where each object is characterized by two different membership functions: a true membership function and a false membership function.
This kind of reasoning is also called interval membership, as opposed to point membership in the context of fuzzy sets.
Mathematical definition.
A vague set formula_1 is characterized by
The "grade of membership" for x is not a crisp value anymore, but can be located in formula_5. This interval can be interpreted as an extension to the fuzzy membership function. The vague set degrades to a fuzzy set, if formula_6 for all x.
The "uncertainty" of x is the difference between the upper and lower bounds of the membership interval; it can be computed as formula_7.