δ -equalities of neutrosophic sets

Fuzzy sets and intuitionistic fuzzy sets can't handle imprecise, indeterminate, inconsistent, and incomplete information. Neutrosophic sets play an important role to overcome this difficulty. A neutrosophic set has a truth membership function, indeterminate membership function, and a falsehood membership functions that can handle all types of ambiguous information. New type of union and intersection has been proposed in this paper. In this paper, δ -equalities of neutrosophic sets have been introduced. Further, some basic properties of δ - equalities have been discussed. Moreover, these δ - equalities have been applied to set theoretic operations of neutrosophic set such as union, intersection, complement, product, probabilistic sum, bold sum, bold intersection, bounded difference, symmetrical difference, and convex linear sum of min and max. These δ -equalities of neutrosophic sets have been further extended to neutrosophic relations and neutrosophic norms respectively. In this paper, δ -equalities also applied in the composition of neutrosophic relations, Cartesian product and neutrosophic triangular norms. The applications and utilizations of δ -equalities have been presented in this paper. In this regards, δ -equalities have been successfully applied in Fault Tree Analysis and Neutrosophic Reliability (generalization of Profust Reliability).