The weak inverse image function associated to $R$1 is the function
\[ R^{-1}\colon \mathcal{P}\webleft (B\webright )\to \mathcal{P}\webleft (A\webright ) \]
defined by2
\[ R^{-1}\webleft (V\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ a\in A\ \middle |\ R\webleft (a\webright )\cap V\neq \text{Ø}\webright\} \]
for each $V\in \mathcal{P}\webleft (B\webright )$.
1Further Terminology: Also called simply the inverse image function associated to $R$.
2Further Terminology: The set $R^{-1}\webleft (V\webright )$ is called the weak inverse image of $V$ by $R$ or simply the inverse image of $V$ by $R$.
