# Definition:Mapping/Definition 2

## Definition

Let $S$ and $T$ be sets.

A mapping $f$ from $S$ to $T$, denoted $f: S \to T$, is a relation $f = \struct {S, T, G}$, where $G \subseteq S \times T$, such that:

$\forall x \in S: \forall y_1, y_2 \in T: \tuple {x, y_1} \in G \land \tuple {x, y_2} \in G \implies y_1 = y_2$

and

$\forall x \in S: \exists y \in T: \tuple {x, y} \in G$

## Notation

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

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 $y$.

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

$x f = y$, as seen in Nathan Jacobson: Lectures in Abstract Algebra: 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: 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.

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

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.

## Also see

• Results about mappings can be found here.