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 $$V$$ is characterized by
 * its true membership function $$t_v(x)$$
 * its false membership function $$f_v(x)$$
 * with $$0 \le t_v(x)+f_v(x) \le 1$$

The grade of membership for x is not a crisp value anymore, but can be located in $$[t_v(x), 1-f_v(x)]$$. This interval can be interpreted as an extension to the fuzzy membership function. The vague set degenerates to a fuzzy set, if $$1-f_v(x)=t_v(x)$$ 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 $$(1-f_v(x))-t_v(x)$$.