I am currently studying for my first exam as a maths student, and is stuck on the problem below. I understood the solution when I saw it, but I am not quite sure of how to explain it properly. If anyone might be able to explain the solution, feel free to do so!
Problem
It is possible to represent a relation $R$ from $A$ to $B$ as a function $f_{R}(a) : A \rightarrow P(B),$ where $P(B)$ is the power set of $B$.
Solution
Let $f_{R}(a) = \{b \in B : aRb\} \in P(B)$ for $a \in A$.
You then have that $aRb \iff b \in f_{R}(a)$ which is a suitable representation of the relation.
I'm not sure what you mean by an "explanation".
The idea here is that you know the relation when for each particular $a \in A$ you know all the elements of $B$ that $a$ is related to. That's the subset of $B$ you are calling $f_R(a)$.