Definition:Projection (Mapping Theory)/First Projection

Definition
Let $S$ and $T$ be sets.

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

The first projection on $S \times T$ is the mapping $\pr_1: S \times T \to S$ defined by:
 * $\forall \tuple {x, y} \in S \times T: \map {\pr_1} {x, y} = x$

Also known as
This is sometimes referred to as:
 * the projection on the first co-ordinate
 * the projection onto the first component

or similar.

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:
 * $\paren x_0 = a$

which is interpreted to mean the same as:
 * $\map {\pr_1} {a, b} = a$

We also have:
 * $\map {\pi^1} {a, b} = a$

On, to avoid all such confusion, the notation $\map {\pr_1} {x, y} = x$ is to be used throughout.

Also see

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