 It follows immediately from  Definition 3.3  that \(\leq_m\) is reflexive. It is also clearly transitive. (For if \(f(x)\) and \(g(x)\) are computable functions which respectively serve as many-one reductions showing \(A \leq_m B\) and \(B \leq_m C\), then their composition \(f(g(x))\) is a many-one reduction showing \(A \leq_m C\).) As it thus follows that \(\equiv_m\) is an equivalence relation, it also makes sense to define the following: