Definition:Restriction/Mapping

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

Let $X \subseteq S$.

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

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


 * $f {\restriction_{X \times Y}} = f \cap \left({X \times Y}\right)$

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:


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

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

Technical Note
The expression:


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

is produced by the following $\LaTeX$ code:

f {\restriction_{X \times Y}}: X \to Y