Definition:Projection (Mapping Theory)

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This page is about projection mappings pertaining to finite Cartesian products. For other uses, see Definition:Projection.

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 $\pr_j: S \to S_j$ defined by:

$\pr_j \left({s_1, s_2, \ldots, s_j, \ldots, s_n}\right) = s_j$

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


Family of Sets

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 $\pr_j: S \to S_j$ defined by:

$\pr_j \left({\left\langle{s_i}\right\rangle_{i \mathop \in I}}\right) = s_j$

where $\left\langle{s_i}\right\rangle_{i \mathop \in I}$ is an arbitrary element of $\displaystyle \prod_{i \mathop \in I} S_i$.


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 $T$.


First Projection

The first projection on $S \times T$ is the mapping $\operatorname{pr}_1: S \times T \to S$ defined by:

$\forall \left({x, y}\right) \in S \times T: \operatorname{pr}_1 \left({x, y}\right) = x$


Second Projection

The second projection on $S \times T$ is the mapping $\operatorname{pr}_2: S \times T \to T$ defined by:

$\forall \left({x, y}\right) \in S \times T: \operatorname{pr}_2 \left({x, y}\right) = y$


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:

$\pr_j \left({a_0, a_1, \ldots, a_{n - 1}}\right) = a_{j - 1}$


On $\mathsf{Pr} \infty \mathsf{fWiki}$, 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

  • Results about projections can be found here.


Sources