Definition:Mapping/Definition 1

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


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 $\cos x$

but instead should say:

the function $\cos$

or:

the function $x \mapsto \cos x$

although for the latter it would be better to also specify the domain and codomain.


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.


Sources

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