Definition:Projection (Mapping Theory)/Family of Sets

Definition
Let $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ be a family of sets.

Let $\displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\left\langle{S_i}\right\rangle_{i \mathop \in I}$.

For each $j \in I$, the $j$th projection on $\displaystyle S = \prod_{i \mathop \in I} S_i$ is the mapping $\operatorname{pr}_j: S \to S_j$ defined by:
 * $\operatorname{pr}_j \left({\left\langle{s_i}\right\rangle_{i \mathop \in I}}\right) = s_j$

where $\left\langle{s_i}\right\rangle_{i \mathop \in I}$ 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.