Definition:Restriction/Relation

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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 \(s \mathrel {\mathcal R {\restriction_{X \times Y} } } t\) is s \mathrel {\mathcal R {\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