I have a question asking me to transform the relation to a total function, how this could be, also the power of set become to be a many valued function ?
Show how a relation $R \subseteq A \times B$ can be transformed into an "equivalent'' total function $f_R: A \rightarrow P(B)$, where $P(B)$ is the power set of $B$. (A function like $f_R$ is sometimes called a many-valued function.)
The codomain $P(B)$ of the total function $f_R$ is different than the codomain $B$ of the relation $R$.
Simply assign $f_R(a):=\{b\in B:(a,b)\in R\}$.
It's not the power set, but a function with values in $P(B)$, like $f_R$, which might be called many-valued function, as it assigns (zero, one or) multiple elements of $B$ to a single element of $A$.
If such a function $f:A\to P(B)$ is given, it also gives rise to a binary relation $R_f$ so that these operations $R\mapsto f_R$ and $f\mapsto R_f$ are inverses to each other.
Can you define $R_f$ in terms of $f$?