 The set of primitive recursive relations is closed under boolean operations. In other words, if \(P(\vec{x})\) and \(Q(\vec{x})\) are primitive recursive, then so are \(\neg P(\vec{x})\), \(P(\vec{x}) \wedge Q(\vec{x})\), \(P(\vec{x}) \vee Q(\vec{x})\), \(P(\vec{x}) \rightarrow Q(\vec{x}),\) and \(P(\vec{x}) \leftrightarrow Q(\vec{x})\).