# Definition:Mapping/Definition 1

## Definition

Let $S$ and $T$ be sets.

A mapping from $S$ to $T$ is a binary relation on $S \times T$ which associates each element of $S$ with exactly one element of $T$.

## Notation

Let $f$ be a mapping.

This is usually denoted $f: S \to T$, which is interpreted to mean:

$f$ is a mapping with domain $S$ and codomain $T$
$f$ is a mapping of (or from) $S$ to (or into) $T$
$f$ maps $S$ to (or into) $T$.

The notation $S \stackrel f {\longrightarrow} T$ is also seen.

For $x \in S, y \in T$, the usual notation is:

$f: S \to T: \map f s = y$

where $\map f s = y$ is interpreted to mean $\tuple {x, y} \in f$.

It is read $f$ of $x$ equals $y$.

This is the preferred notation on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also defined as

Some sources do not place emphasis on the uniqueness of the element of $T$ that is being mapped to.

## Also see

• Results about mappings can be found here.

## Technical Note

The $\LaTeX$ code for $\map {f} {x}$ is \map {f} {x} .

When the argument is a single character, it is usual to omit the braces:

\map f x