Definition:Extension of Relation

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 $$\mathcal{R}_1$$.