Definition:Restriction/Relation

Definition

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:

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

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

• Extension of a Relation

• Restriction of a Mapping
• Restriction of an Operation

• Properties of Restriction of Relation‎

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 14$