Definition:Restriction/Relation

Definition
Let $\RR$ be a relation on $S \times T$.

Let $X \subseteq S$, $Y \subseteq T$.

The restriction of $\RR$ to $X \times Y$ is the relation on $X \times Y$ defined as:


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

Note that the parenthesis is not necessary in the above, but it does make the meaning clearer.

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

A different way of saying the same thing is:


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

Also known as
Some sources refer to $\RR {\restriction_X}$ as the relation induced on $X$ by $\RR$.

Also see

 * Definition:Extension of Relation


 * Definition:Restriction of Mapping
 * Definition:Restriction of Operation


 * Properties of Restriction of Relation‎