Let $R$ be a binary relation. If $A\subseteq\dom R$, the left-restriction / restriction of $R$ to $A$ is
\[{R\restriction_A} = \{(x,y):x\,R\,y\land x\in A\}.\]If $B\subseteq\ran R$, the right-restriction of $R$ to $B$ is
\[{R\restriction^B} = \{(x,y):x\,R\,y\land y\in B\}.\]Both $R\restriction_A$ and $R\restriction^B$ are sets:
\[{R\restriction_A}\subseteq R, \quad {R\restriction^B}\subseteq R.\]
For any binary relation $R$,
\[{R\restriction_\varnothing} = {R\restriction^\varnothing} = \varnothing.\]