# Definition:Restriction

## Contents

## Definition

### Restriction of a Relation

Let $\mathcal R$ be a relation on $S \times T$.

Let $X \subseteq S$, $Y \subseteq T$.

The **restriction of $\mathcal R$ to $X \times Y$** is the relation on $X \times Y$ defined as:

- $\mathcal R {\restriction_{X \times Y} }: = \mathcal R \cap \paren {X \times Y}$

If $Y = T$, then we simply call this the **restriction of $\mathcal R$ to $X$**, and denote it as $\mathcal R {\restriction_X}$.

### Restriction of a Mapping

Let $f: S \to T$ be a mapping.

Let $X \subseteq S$.

Let $f \sqbrk X \subseteq Y \subseteq T$.

The **restriction of $f$ to $X \times Y$** is the mapping $f {\restriction_{X \times Y} }: X \to Y$ defined as:

- $f {\restriction_{X \times Y} } = f \cap \paren {X \times Y}$

If $Y = T$, then we simply call this the **restriction of $f$ to $X$**, and denote it as $f {\restriction_X}$.

### Restriction of an Operation

In the same way that a restriction is defined on a relation, it can be defined on a binary operation.

Let $\left({S, \circ}\right)$ be an algebraic structure, and let $T \subseteq S$.

The **restriction of $\circ$ to $T \times T$** is denoted $\circ {\restriction_T}$, and is defined as:

- $\forall t_1, t_2 \in T: t_1 \mathbin{\circ {\restriction_T}} t_2 = t_1 \circ t_2$

The notation $\circ {\restriction_T}$ is generally used only if it is necessary to emphasise that $\circ {\restriction_T}$ is strictly different from $\circ$ (through having a different domain). When no confusion is likely to result, $\circ$ is generally used for both.

Thus in this context, $\left({T, \circ {\restriction_T}}\right)$ and $\left({T, \circ}\right)$ mean the same thing.

## Notation

The use of the symbol $\restriction$ is a recent innovation over the more commonly-encountered $|$.

Thus the notation $\mathcal R |_{X \times Y}$ and $\struct {T, \circ|_T}$, etc. are currently more likely to be seen than $\mathcal R {\restriction_{X \times Y} }$ and $\struct {T, \circ {\restriction_T} }$.

No doubt as the convention becomes more established, $\restriction$ will develop.

It is strongly arguable that $\restriction$, affectionately known as the **harpoon**, is preferable to $|$ as the latter is suffering from the potential ambiguity of overuse.

Some authors prefer not to subscript the subset, and render the notation as:

- $f \mathbin \restriction X = \set {\tuple {x, \map f x}: x \in X}$

but this is not recommended on $\mathsf{Pr} \infty \mathsf{fWiki}$ because it has less clarity.

## Also see

## Technical Note

The $\LaTeX$ code for \(f {\restriction_{X \times Y} }: X \to Y\) is `f {\restriction_{X \times Y} }: X \to Y`

.

Note that because of the way MathJax renders the image, the restriction symbol and its subscript `\restriction_T`

need to be enclosed within braces `{ ... }`

in order for the spacing to be correct.

The $\LaTeX$ code for \(s \mathrel {\mathcal R {\restriction_{X \times Y} } } t\) is `s \mathrel {\mathcal R {\restriction_{X \times Y} } } t`

.

The $\LaTeX$ code for \(t_1 \mathbin {\circ {\restriction_T} } t_2\) is `t_1 \mathbin {\circ {\restriction_T} } t_2`

.

Again, note the use of `\mathrel { ... }`

and `\mathbin { ... }`

so as to render the spacing evenly.