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.¹ 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 ${\displaystyle V}$ is characterized by

• its true membership function ${\displaystyle t_{v}(x)}$
• its false membership function ${\displaystyle f_{v}(x)}$
• with ${\displaystyle 0\leq t_{v}(x)+f_{v}(x)\leq 1}$

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