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:


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

Also see

 * Extension of a Relation


 * Restriction of a Mapping
 * Restriction of an Operation