Vague set

In mathematics, vague sets are an extension of fuzzy sets. In mathematics, vague sets are an extension of fuzzy sets. In a fuzzy set, each object is assigned a single value in the interval 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 V {displaystyle V} is characterized by The grade of membership for x is not a crisp value anymore, but can be located in [ t v ( x ) , 1 − f v ( x ) ] {displaystyle } . This interval can be interpreted as an extension to the fuzzy membership function. The vague set degrades to a fuzzy set, if 1 − f v ( x ) = t v ( x ) {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 ( 1 − f v ( x ) ) − t v ( x ) {displaystyle (1-f_{v}(x))-t_{v}(x)} .