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 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 is characterized by
The grade of membership for x is not a crisp value anymore, but can be located in. This interval can be interpreted as an extension to the fuzzy membership function. The vague set degrades to a fuzzy set, if 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.