Definition:Projection (Mapping Theory)

Mappings
Let $S$ and $T$ be sets.

First Projection
The first projection on $S \times T$ is the mapping $\operatorname{pr}_1: S \times T \to S$ defined by:
 * $\forall \left({x, y}\right) \in S \times T: \operatorname{pr}_1 \left({x, y}\right) = x$

This is sometimes referred to as the projection on the first co-ordinate.

Second Projection
The second projection on $S \times T$ is the mapping $\operatorname{pr}_2: S \times T \to T$ defined by:
 * $\forall \left({x, y}\right) \in S \times T: \operatorname{pr}_2 \left({x, y}\right) = y$

This is sometimes referred to as the projection on the second co-ordinate.

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

For each $j \in \left[{1 \,. \, . \, n}\right]$, the $j$th projection on $\displaystyle S = \prod_{i=1}^n S_i$ 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 $\left({s_1, \ldots, s_n}\right) \in S$.

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

Coordinate Geometry
Let $M$ and $N$ be distinct lines through the plane through the origin.

The projection on $M$ along $N$ is the mapping $\operatorname{pr}_{M, N}$ such that:
 * $\forall x \in \R^2: \operatorname{pr}_{M, N} \left({x}\right) =$ the intersection of $M$ with the line through $x$ parallel to $N$.