\begin{definition}[Definition:Projection (Mapping Theory)/Family of Sets]
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets.
Let $\ds \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$.
For each $j \in I$, the '''$j$th projection on $\ds S = \prod_{i \mathop \in I} S_i$''' is the mapping $\pr_j: S \to S_j$ defined by:
:$\map {\pr_j} {\family {s_i}_{i \mathop \in I} } = s_j$
where $\family {s_i}_{i \mathop \in I}$ is an arbitrary element of $\ds \prod_{i \mathop \in I} S_i$.
\end{definition}