# Definition:Restriction/Relation

## Definition

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 $\mathcal R {\restriction_X}$.

A different way of saying the same thing is:

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

## 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 $\mathcal R {\restriction_X}$ as the **relation induced on $X$ by $\mathcal R$**.

## 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 \(s \mathrel {\RR {\restriction_{X \times Y} } } t\) 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.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Relations - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 14$ - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.19$: Some Important Properties of Relations - 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 6.6$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations: Exercise $3.2$