# Definition:Mapping/Notation

## Notation for Mapping

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 $\mathsf{Pr} \infty \mathsf{fWiki}$.

Sometimes the brackets are omitted: $f x = y$, as seen in Allan Clark: Elements of Abstract Algebra, for example.

The notation $f: x \mapsto y$ is often seen, read $f$ maps, or sends, $x$ to $y$.

In the context of index families, the conventional notation $x_i$ is used to denote the value of the index $i$ under the indexing function $x$.

Thus $x_i$ means the same thing as $\map x i$.

Some sources use this convention for the general mapping, thus:

$f_x = y$

as remarked on in P.M. Cohn: Algebra Volume 1 (2nd ed.), for example.

Less common notational forms of $\map f s = y$ are:

$x f = y$, as seen in Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts and 1968: Ian D. Macdonald: The Theory of Groups, for example
$x^f = y$, as seen in Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts and John D. Dixon: Problems in Group Theory, for example
This left-to-right style is referred to by some authors as the European convention.

John L. Kelley: General Topology provides a list of several different styles: $\tuple {f, x}$, $\tuple {x, f}$, $f x$, $x f$ and $\cdot f x$, and discusses the advantages and disadvantages of each.

The notation $\cdot f x$ is attributed to Anthony Perry Morse, and can be used to express complicated expressions without the need of parenthesis to avoid ambiguity. However, it appears not to have caught on.

### Warning

The notation:

Let $\map f x$ be a mapping (or function) ...

is an abuse of notation.

If $f: S \to T$ is a mapping, then $\map f x \in T$ for all $x \in S$.

Thus $\map f x$ is a mapping if and only if $\Img f$ is a set of mappings.

The point is that, as used here, $\map f x$ is not a mapping, but it is the image of $x$ under $f$.

Hence it is preferable not to talk about:

the function $\cos x$

the function $\cos$
the function $x \mapsto \cos x$