 In order to appreciate this passage, it is again useful to think of a set \(A \subseteq \mathbb{N}\) as being associated with the problem of deciding membership in \(A\)—e.g., given a natural number \(n\), is \(n\) prime? (i.e., \(n \in \textit{PRIMES}\)?) or is the \(n\)th partial computable function with input \(n\) defined? (i.e., \(n \in K\)?). But even given this correspondence, the assertion that a solution to a problem \(B\) “immediately yields” a solution to \(A\) may still be analyzed in a number of different ways. Two of the most important possibilities are as follows: