Definition 1. Let $R$ be a binary relation. The image of a set $A$ under $R$ is
\[R[A] = \{y : (\exists x \in A : x \,R\, y)\}.\]This is a set by Separation Schema:
\[R[A] \subseteq \im R.\]Definition 2. Let $R$ be a binary relation. The preimage of a set $B$ under $R$ is $R^{-1}[B]$, the image of $B$ under the converse $R^{-1}$.
Note. Preimage is also known as inverse image.