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.
Mathematical definition
A vague set [math]\displaystyle{ V }[/math] is characterized by
- its true membership function [math]\displaystyle{ t_v(x) }[/math]
- its false membership function [math]\displaystyle{ f_v(x) }[/math]
- with [math]\displaystyle{ 0 \le t_v(x)+f_v(x) \le 1 }[/math]
The grade of membership for x is not a crisp value anymore, but can be located in [math]\displaystyle{ [t_v(x), 1-f_v(x)] }[/math]. This interval can be interpreted as an extension to the fuzzy membership function. The vague set degenerates to a fuzzy set, if [math]\displaystyle{ 1-f_v(x)=t_v(x) }[/math] 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 [math]\displaystyle{ (1-f_v(x))-t_v(x) }[/math].
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