Vague set

{{Short description|System in mathematical set theory}}

{{more citations needed|date=December 2012}}

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.[https://ieeexplore.ieee.org/document/229476/ "Vague sets"]. 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).

See also

References

{{reflist}}