Definition 1. Let $R$ be a binary relation. The (left-)restriction of $R$ to a set $A$ is
\[{R \restriction_A} = \{(x, y) : x \,R\, y \,\land\, x \in A\}.\]The right-restriction of $R$ to a set $B$ is
\[{R \restriction^B} = \{(x, y) : x \,R\, y \,\land\, y \in B\}.\]These are sets by Separation Schema:
\[{R \restriction_A} \subseteq R, \quad {R \restriction^B} \subseteq R.\]