# Definition:Extension of Relation

## Definition

Let:

$\mathcal R_1 \subseteq X \times Y$ be a relation on $X \times Y$
$\mathcal R_2 \subseteq S \times T$ be a relation on $S \times T$
$X \subseteq S$
$Y \subseteq T$
$\mathcal R_2 \restriction_{X \times Y}$ be the restriction of $\mathcal R_2$ to $X \times Y$.

Let $\mathcal R_2 \restriction_{X \times Y} = \mathcal R_1$.

Then $\mathcal R_2$ extends or is an extension of $\mathcal R_1$.