Definition:Restriction/Mapping

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Definition

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

Let $X \subseteq S$.

Let $\operatorname{Im} \left({X}\right) \subseteq Y \subseteq T$.


The restriction of $f$ to $X \times Y$ is defined as:

$f \restriction_{X \times Y}: X \to Y = f \cap X \times Y$


If the codomain of $f \restriction_{X \times Y}$ is understood to be $\operatorname{Cdm} \left({f}\right)$, i.e. $Y = \operatorname{Cdm} \left({f}\right)$, then we define the restriction of $f$ to $X$ as:

$f \restriction_X: X \to \operatorname{Cdm} \left({f}\right) = f \cap X \times \operatorname{Cdm} \left({f}\right)$


A different way of saying the same thing is:

$f \restriction_X = \left\{{\left({x, y}\right) \in f: x \in X}\right\}$

or:

$f \restriction_X = \left\{{\left({x, f \left({x}\right)}\right): x \in X}\right\}$


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 $|$.

Thus the notation $\mathcal R |_{X \times Y}$ and $\left({T, \circ|_T}\right)$, etc. are currently more likely to be seen than $\mathcal R \restriction_{X \times Y}$ and $\left({T, \circ \restriction_T}\right)$.

No doubt as the convention becomes more established, $\restriction$ will develop.

It is strongly arguable that $\restriction$, affectionately known as 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:

$f \restriction X = \left\{{\left({x, f \left({x}\right)}\right): x \in X}\right\}$


Also see


Sources

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