  For \(S \subseteq \mathbf{N}\), the complement of \(S\) is \(\mathbf{N} - S\), i.e., the set of all natural numbers not in \(S\). We say that the set \(S\) is co-r.e. if and only if its complement is r.e. If a set, \(S\), is r.e. and co-r.e. then we can list out all of its elements in one column and we can list out all of its non-elements in a second column. In this way we can decide whether or not a given element, \(n\), is in \(S\): just scan the two columns and wait for \(n\) to show up. If it shows up in the first column then \(n \in S\). Otherwise it will show up in the second column and \(n \not\in S\). In fact, a set is recursive iff it is r.e. and co-r.e.