Definition:Projection (Mapping Theory)/Second Projection

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

Also see

 * Definition:First Projection


 * Definition:Right Operation: the same concept in the context of abstract algebra.