Definition:Mapping/Notation

Notation for Mapping
Let $f = \tuple {S, T, R}$, 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:


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

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.

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 $y$.

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, for example.

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


 * $x f = y$, as seen in and, for example
 * $x^f = y$, as seen in and, for example
 * This left-to-right style is referred to by some authors as the European convention.

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, and can be used to express complicated expressions without the need of parenthesis to avoid ambiguity. However, it appears not to have caught on.