Definition:Restriction

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:


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

An alternative 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:


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

An alternative 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 more likely to be seen at the moment 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

 * Extension of a Relation
 * Extension of a Mapping
 * Extension of an Operation