Definition 1. Let $R$ be a binary relation.
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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:
\[{R \restriction_A} \subseteq R, \quad {R \restriction^B} \subseteq R.\]