Definition:Restriction/Mapping

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

Let $X \subseteq S$.

Let $f \sqbrk X \subseteq Y \subseteq T$.

The restriction of $f$ to $X \times Y$ is the mapping $f {\restriction_{X \times Y} }: X \to Y$ defined as:


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

If $Y = T$, then we simply call this the restriction of $f$ to $X$, and denote it as $f {\restriction_X}$.

A different way of saying the same thing is:


 * $f {\restriction_X} = \set {\tuple {x, y} \in f: x \in X}$

or:
 * $f {\restriction_X} = \set {\tuple {x, f \paren x}: x \in X}$

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

 * Definition:Extension of Mapping


 * Definition:Injective Restriction
 * Definition:Surjective Restriction
 * Definition:Bijective Restriction


 * Definition:Restriction of Relation
 * Definition:Restriction of Operation