\begin{definition}[Definition:Restriction/Relation]
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:
:$\RR {\restriction_{X \times Y} }: = \RR \cap \paren {X \times Y}$
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:
:$\RR {\restriction_X} = \set {\tuple {x, y} \in \RR: x \in X}$
\end{definition}