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 $$\mathcal R_1$$.

Also see

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