Entropy for Vague sets based on convex simplex

To the drawbacks in the existed methods for constructing the entropy of Vague sets, a method on the basis of convex simplex is proposed. By use of a corresponding relation between Vague sets and 3 triangles within a same plane of a convex simplex, a geometrical expression for Vague sets is presented, and thus the quantities of the degree of true membership, false membership and unknown for Vague sets and their contrastive relations between them are expressed intuitively and visually. Then an entropy for Vague sets is proposed and the corresponding geometrical interpretation is given. Examples show that, one, the entropy of Vague sets is the same as that of Fuzzy sets if Vague sets turn into Fuzzy sets; two, the entropy of Vague sets is monotone in subsections if the degree of unknown is the same between Vague sets; three, the entropy for Vague sets has extremum property and symmetry property if the degree of true membership is equal the degree of false membership. The proposed method solves the problems in the existed methods for constructing the entropy of Vague sets, and is a correct, reasonable and effective method.