# Definition:Projection (Mapping Theory)/Family of Sets

## Definition

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets.

Let $\displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$.

For each $j \in I$, the $j$th projection on $\displaystyle S = \prod_{i \mathop \in I} S_i$ 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 $\displaystyle \prod_{i \mathop \in I} S_i$.

## Also known as

This is sometimes referred to as the projection on the $j$th co-ordinate.