It is easy to see that \(\leq_P\) is a reflexive relation. Since the composition of two polynomial time computable functions is also polynomial time computable, \(\leq_P\) is also transitive. We additionally say that a class \(\textbf{C}\) is closed under \(\leq_P\) if \(Y \in \textbf{C}\) and \(X \leq_P Y\) implies \(X \in \textbf{C}\). It is also easy to see that the classes \(\textbf{P}, \textbf{NP}, \textbf{PSPACE},\textbf{EXP}, \textbf{NEXP}\) and \(\textbf{EXPSPACE}\) are closed under this relation.[16] A problem \(Y\) is said to be hard for a class \(\textbf{C}\) if \(X \leq_P Y\) for all \(X \in \textbf{C}\). Finally \(Y\) is said to be complete for \(\textbf{C}\) if it is both hard for \(\textbf{C}\) and also a member of \(\textbf{C}\).