# Definition:Mapping/Notation

## Notation for Mapping

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**.

- 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 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.

## Historical Note

The notation $\map f x$ was introduced by Leonhard Paul Euler.

## Sources

- 1945: A. Geary, H.V. Lowry and H.A. Hayden:
*Advanced Mathematics for Technical Students, Part I*... (previous) ... (next): Chapter $\text I$: Differentiation: Functional notation - 1947: James M. Hyslop:
*Infinite Series*(3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 4$: Limits of Functions (footnote $\ddagger$) - 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Functions - 1963: Morris Tenenbaum and Harry Pollard:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text B$: The Meaning of the Term*Function of One Independent Variable*: Definition $2.4$ - 1967: John D. Dixon:
*Problems in Group Theory*... (previous) ... (next): Introduction: Notation - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Mappings: $\S 10$ - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 4$. Relations; functional relations; mappings:*Remark $3$* - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 20$: Introduction: Remarks $\text{(f)}$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**function (map, mapping)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**function (map, mapping)** - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 9$ Functions

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