Definition:Mapping/Definition 4

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Definition

Let $S$ and $T$ be sets.

A mapping from $S$ to $T$ is a relation on $S \times T$ which is:

$(1): \quad$ Many-to-one
$(2): \quad$ Left-total, that is, defined for all elements in $S$.


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.


Sources