# Definition:Restriction/Relation

## Definition

Let $\RR$ be a relation on $S \times T$.

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

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

$\RR {\restriction_{X \times Y} }: = \RR \cap \paren {X \times Y}$

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

A different way of saying the same thing is:

$\RR {\restriction_X} = \left\{{\left({x, y}\right) \in \RR: x \in X}\right\}$

## 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 known as

Some sources refer to $\RR {\restriction_X}$ as the relation induced on $X$ by $\RR$.

## 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 $s \mathrel {\RR {\restriction_{X \times Y} } } t$ is s \mathrel {\RR {\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.