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

Projection from Product of Two 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 $\operatorname{pr}$.

Some sources, in particular those which approach the subject from the context of physics, use a superscript: $\operatorname{pr}^j$.

Some sources use the notation $p_j$ for $\operatorname{pr}$.

Also known as
This is sometimes referred to as the projection on the $j$th co-ordinate, and hence generically as a coordinate projection.

Some sources use a $0$-based system to number the elements of a Cartesian product.

For a given ordered $n$-tuple $x = \left({a_0, a_1, \ldots, a_{n-1}}\right)$, the notation $\left({x}\right)_j$ is also seen.

Hence:
 * $\left({x}\right)_j = a_j$

which is interpreted to mean the same as:
 * $\operatorname{pr}_j \left({a_0, a_1, \ldots, a_{n-1}}\right) = a_{j-1}$

On, to avoid all such confusion, the notation $\operatorname{pr}_j \left({a_1, a_2, \ldots, a_n}\right) = a_j$ is to be used throughout.

Also see

 * Projections are Surjections
 * Projections are Epimorphisms


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