# Definition:Mapping/Definition 4

## Contents

## 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 $y$**.

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: 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**.

- This left-to-right style is referred to by some authors as the

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:

is inaccurate and misleading.

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.

Hence we should not 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 see

- Results about
**mappings**can be found here.

## Sources

- 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 4$. Relations; functional relations; mappings - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions