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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:

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


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.