Definition:Projection (Mapping Theory)

Definition
Let $S_1, S_2, \ldots, S_j, \ldots, S_n$ be sets.

Let $\displaystyle \prod_{i \mathop = 1}^n S_i$ be the Cartesian product of $S_1, S_2, \ldots, S_n$.

For each $j \in \left\{{1, \ldots, n}\right\}$, the $j$th projection on $\displaystyle S = \prod_{i \mathop = 1}^n S_i$ is the mapping $\operatorname{pr}_j: S \to S_j$ defined by:
 * $\operatorname{pr}_j \left({s_1, s_2, \ldots, s_j, \ldots, s_n}\right) = s_j$

for all $\left({s_1, \ldots, s_n}\right) \in S$.

Family of Sets
The definition is most usually seen in the context of the Cartesian product of two sets, as follows.

Let $S$ and $T$ be sets.

Let $S \times T$ be the Cartesian product of $S$ and $T$.

Also denoted as
It is common to denote projections with the Greek letter $\pi$ (pi) in place of $\operatorname{pr}$.

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

Also see

 * Projections are Surjections
 * Projections are Epimorphisms


 * The left operation and right operation for the same concept in the context of abstract algebra.