Let $S$ and $T$ be two sets. If $R$ is a relation of $S \times T$, then for each $t \in T$we have the pre-image
$_{R}{[t]}=\{s \in S| sRt\} $
Which is a subset of $S$
Prove that the relation $\{(t,_{R}[t])|t \in T\}$ is a map from $T$ to the power set $\mathscr{P}(S)$ of $S$.
Moreover, show that, if $f:T \rightarrow \mathscr{P}(S)$is a map, then $R_{f} = \{(s,t)|s \in f(t)\}$is a relation on $S \times T$ with $_{R_{f}}\textrm{[t]}=f$
I am not sure how to even start. Do I have to first show that the relation is a function first or is it obvious from the definition?