# Definition:Restriction/Relation/Class Theory

## Definition

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:

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

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

## 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

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering