True-False Sets

In this paper, we introduce the notion of True-False sets or briefly TF-sets, as a generalization of fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets, neutrosophic sets and vague sets and we investigate some related properties. By considering the concept of α-cuts, we state and prove decomposition theorem and representation theorem for TF-sets. Then we define the preference relations on TF-sets by using two matrices, true matrix and false matrix and investigate some related properties of Them. Finally, we solve a decision making problem by using preference TF-relations.