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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.
A vague set is characterized by
- its true membership function
- its false membership function
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 .