Some existing theories of triangular intuitionistic fuzzy numbers (TIFNs) and triangular intuitionistic fuzzy sets (TIFSs) are proved in this paper. In addition, this paper also provides some novel theorems and related proofs on TIFNs and TIFSs. Moreover, the presented theories are applied in the field of two-sided matching decision-making. A numerical case is adopted for demonstrating the effectiveness of the proposed theories and method. This work will effectively complement the theories of TIFNs and TIFSs, and popularize their application scope.
