\begin{definition}[Definition:Projection (Mapping Theory)]
Let $S_1, S_2, \ldots, S_j, \ldots, S_n$ be sets.
Let $\ds \prod_{i \mathop = 1}^n S_i$ be the Cartesian product of $S_1, S_2, \ldots, S_n$.
For each $j \in \set {1, 2, \ldots, n}$, the '''$j$th projection on $\ds S = \prod_{i \mathop = 1}^n S_i$''' is the mapping $\pr_j: S \to S_j$ defined by:
:$\map {\pr_j} {s_1, s_2, \ldots, s_j, \ldots, s_n} = s_j$
for all $\tuple {s_1, s_2, \ldots, s_n} \in S$.
\end{definition}