 Given \(A \subseteq \mathbb{N}\), we may also consider \(K^A =\{n : \phi^{A}_n(n) \darrow\}\)—i.e., the set considered above which corresponds to the Diagonal Halting Problem relativized to the oracle \(A\). \(K^A\) is referred to as the jump of \(A\) for which we also write \(A'\). This notation is also used to denote an operation on Turing degrees by setting \(\mathbf{a}' = \textrm{deg}_T(A')\) for some representative \(A \in \mathbf{a}\). The following collects together several facts about the jump operation on both sets and degrees: