# Definition:Mapping/Definition 2

## Contents

## 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 $y$**.

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

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: I. Basic Concepts*... (previous) ... (next): Introduction $\S 2$: Product sets, mappings - 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Functions - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 8$: Functions - 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.3$: Definition $1.8$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $1$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.4$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Functions - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.10$: Functions: Definition $10.1$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 20$: Introduction: Remarks $\text {(k)}$ - 2011: Robert G. Bartle and Donald R. Sherbert:
*Introduction to Real Analysis*(4th ed.) ... (previous) ... (next): $\S 1.1$: Sets and Functions