Definition:Restriction
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Definition
Restriction of a Relation
Let $\mathcal R$ be a relation on $S \times T$.
Let $X \subseteq S$.
Let $\operatorname{Im} \left({X}\right) \subseteq Y \subseteq T$.
The restriction of $\mathcal R$ to $X \times Y$ is defined as:
- $\mathcal R \restriction_{X \times Y}: X \to Y = \mathcal R \cap X \times Y$
If the codomain of $\mathcal R \restriction_{X \times Y}$ is understood to be $\operatorname{Cdm} \left({\mathcal R}\right)$, i.e.
- $Y = \operatorname{Cdm} \left({\mathcal R}\right)$
then we define the restriction of $\mathcal R$ to $X$ as:
- $\mathcal R \restriction_X: X \to \operatorname{Cdm} \left({\mathcal R}\right) = \mathcal R \cap X \times \operatorname{Cdm} \left({\mathcal R}\right)$
A different way of saying the same thing is:
- $\mathcal R \restriction_X = \left\{{\left({x, y}\right) \in \mathcal R: x \in X}\right\}$
Restriction of a Mapping
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 $X$.
Restriction of an Operation
In the same way that a restriction is defined on a relation, it can be defined on a binary operation.
Let $\left({S, \circ}\right)$ be an algebraic structure, and let $T \subseteq S$.
The restriction of $\circ$ to $T \times T$ is defined as:
- $\left({T, \circ \restriction_T}\right): t_1, t_2 \in T: t_1 \circ \restriction_T \ t_2 = t_1 \circ t_2$
The notation $\circ \restriction_T$ is generally used only if it is necessary to emphasise that $\circ \restriction_T$ is strictly different from $\circ$ (through having a different domain and codomain). When no confusion is likely to result, $\circ$ is generally used for both.
Thus in this context, $\left({T, \circ \restriction_T}\right)$ and $\left({T, \circ}\right)$ mean the same thing.
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:
- $f \restriction X = \left\{{\left({x, f \left({x}\right)}\right): x \in X}\right\}$
