# Definition:Projection (Mapping Theory)/Second Projection

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## Definition

Let $S$ and $T$ be sets.

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

The second projection on $S \times T$ is the mapping $\pr_2: S \times T \to T$ defined by:

$\forall \tuple {x, y} \in S \times T: \map {\pr_2} {x, y} = y$

## Also known as

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

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

For a given ordered pair $x = \tuple {a, b}$, the notation $\paren x_n$ is also seen.

Hence:

$\paren x_2 = b$

which is interpreted to mean the same as:

$\map {\pr_2} {a, b} = b$

We also have:

$\map {\pi^2} {a, b} = b$

On $\mathsf{Pr} \infty \mathsf{fWiki}$, to avoid all such confusion, the notation $\map {\pr_2} {x, y} = y$ is to be used throughout.