Definition:Projection (Mapping Theory)

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

Definition

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

Let $\displaystyle \prod_{i=1}^n S_i$ 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 $\displaystyle S = \prod_{i=1}^n S_i$ 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 $\left({s_1, \ldots, s_n}\right) \in S$.

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 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.

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 9$: Families
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 18$