# 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:

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

where $R \subseteq S \times T$ is the subset of the Cartesian product of $S$ and $T$ which defines the relation $\RR$.

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:

- $\RR {\restriction_X} = \set {\tuple {x, y} \in R: x \in X}$

Note that the parenthesis is not necessary in the above, but it does make the meaning clearer.

### Class Theory

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a relation in $V$.

Let $A \subseteq V$ be a class.

The **restriction of $\RR$ to $A$** is the relation on $A \times A$ defined as:

- $\RR {\restriction_A}: = \RR \cap \paren {A \times A}$

## Notation

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

Thus the notation $\RR \vert_{X \times Y}$ and $\struct {T, \circ \vert_T}$, etc. are currently more likely to be seen than $\RR {\restriction_{X \times Y} }$ 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 $\vert$ 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 note that it is commonplace even to omit the $\restriction$ symbol altogether, and merely render as $\RR_{X \times Y}$ or $\struct {T, \circ_T}$, and so on.

## Also known as

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

## Also see

- Results about
**restrictions**can be found**here**.

## 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): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings - 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 $3.2$