Definition:Extension of Relation

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 $$\mathcal{R}_1$$.

Mapping
As a mapping is also a relation, the definition applies directly to mappings:

Let:


 * $$f_1 \subseteq X \times Y$$ be a mapping on $$X \times Y$$;
 * $$f_2 \subseteq S \times T$$ be a mapping on $$S \times T$$;
 * $$X \subseteq S$$;
 * $$Y \subseteq T$$;
 * $$f_2 \restriction_{X \times Y}$$ be the restriction of $$f_2$$ to $$X \times Y$$.

Let $$f_2 \restriction_{X \times Y} = f_1$$.

Then $$f_2$$ extends or is an extension of $$f_1$$.