 Definition 2.4  thus conventionalizes the characterization of a primitive recursive relation \(R \subseteq \mathbb{N}^k\) as one for which there exists an algorithm similar to that illustrated above which returns the output 1 on input \(\vec{n}\) if \(R\) holds of \(\vec{n}\) and the output 0 if \(R\) does not hold of \(\vec{n}\). As will become clear below, most sets and relations on the natural numbers which are considered in everyday mathematics—e.g., the set PRIMES of prime numbers or the relation   \[\textit{DIV} = \{\langle n, m \rangle : n \textit{ divides } m \textit{ without remainder}\}\]  —are primitive recursive.