True-False triangular norm and conorms and It’s Application

In this article, we introduced triangular norm T (briefly t-norm) and triangular co-norm S (briefly s-norm) of combination a fuzzy set and an interval-valued fuzzy set and proposed T-representable and S-representable of them. By using T-representable and S-representable, we suggested (T,S)-representable of true-false sets (TFS). We defined the TF implication, the residuated TF implication and considered property between residuated TF implication with Lukasiewicz t-norm and Hamming distance, Hausdorff distance, Chebyshev distance and Moore distance. Also, we defined the degree of similarity between two TFS and considered property between the degree of similarity with some type of distance. Later on, we extended the TOPSIS method based on TFS. Finally, we applied proposed the TOPSIS method solve the decision-making problem.