Definition:Restriction/Mapping

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Definition

Let $f: S \to T$ be a mapping.

Let $X \subseteq S$.

Let $f \sqbrk X \subseteq Y \subseteq T$.


The restriction of $f$ to $X \times Y$ is the mapping $f {\restriction_{X \times Y} }: X \to Y$ defined as:

$f {\restriction_{X \times Y} } = f \cap \paren {X \times Y}$


If $Y = T$, then we simply call this the restriction of $f$ to $X$, and denote it as $f {\restriction_X}$.


A different way of saying the same thing is:

$f {\restriction_X} = \set {\tuple {x, y} \in f: x \in X}$

or:

$f {\restriction_X} = \set {\tuple {x, f \paren x}: x \in X}$


This definition follows directly from that for a relation owing to the fact that a mapping is a special kind of relation.


Note that $f {\restriction_X}$ is a mapping whose domain is $X$.


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.


Examples

Restriction of Square Function on Natural Numbers

Let $f: \N \to \N$ be the mapping defined as:

$\forall n \in \N: \map f n = n^2$

Let $S = \set {x \in \N: \exists y \in \N_{>0}: x = 2 y} = \set {2, 4, 6, 8, \ldots}$

Let $g: S \to \N$ be the mapping defined as:

$\forall n \in \N: \map g n = n^2$

Then $g$ is a restriction of $f$.


Bijective Restriction of Real Sine Function

Let $f: \R \to \R$ be the mapping defined as:

$\forall x \in \R: f \paren x = \sin x$


Then a bijective restriction $g$ of $f$ can be defined as:

$g: S \to T: \forall x \in S: g \paren x = \sin x$

where:

$S = \closedint {-\dfrac \pi 2} {\dfrac \pi 2}$
$T = \closedint {-1} 1$


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

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