Definition:Mapping/Notation

Notation for Mapping
Let $f = \left({S, T, R}\right)$, where $R \subseteq S \times T$, 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:


 * $f: S \to T: f \left({x}\right) = y$

where $f \left({x}\right) = y$ is interpreted to mean $\left({x, y}\right) \in f$.

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

This is the preferred notation on.

Sometimes the brackets are omitted: $f x = y$, as seen in, for example.

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

Less common notational forms of $f \left({x}\right) = y$ are:


 * $x f = y$, as seen in and, for example
 * $x^f = y$, as seen in, for example
 * $f_x = y$, as remarked on in, for example.

provides a list of several different styles: $\left({f, x}\right)$, $\left({x, f}\right)$, $f x$, $x f$ and $\cdot f x$, and discusses the advantages and disadvantages of each..

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