Definition:Restriction/Mapping

Definition
Let $f: S \to T$ be a mapping.

Let $X \subseteq S$.

Let $\operatorname{Im} \left({X}\right) \subseteq Y \subseteq T$.

The restriction of $f$ to $X \times Y$ is defined as:


 * $f \restriction_{X \times Y}: X \to Y = f \cap X \times Y$

If the codomain of $f \restriction_{X \times Y}$ is understood to be $\operatorname{Cdm} \left({f}\right)$, i.e. $Y = \operatorname{Cdm} \left({f}\right)$, then we define the restriction of $f$ to $X$ as:


 * $f \restriction_X: X \to \operatorname{Cdm} \left({f}\right) = f \cap X \times \operatorname{Cdm} \left({f}\right)$

A different way of saying the same thing is:


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

or:
 * $f \restriction_X = \left\{{\left({x, f \left({x}\right)}\right): x \in X}\right\}$

This definition follows directly from that for a relation owing to the fact that a mapping is a special kind of relation.

Note that $f \restriction_X$ is a mapping whose domain is $X$.

Also see

 * Extension of a Mapping


 * Restriction of a Relation
 * Restriction of an Operation