The way I always saw it was that a relation is a subset of $A \times B$, or a collection of ordered pairs $(a,b)$, where $a \in A$ and $b \in B$. Is there any meaningful distinction between the two definitions?
If they are one and the same, I think my trouble lies in the definition of "mapping", and any insight you can give would be great.
The reason I ask this question is because this is how it's defined in my discrete mathematics class.
Thanks for your help!
Yes. If you have any subset $R$ of $A\times B$, then the sections, or cuts, on $A$ define subsets of $B$: $$R_a=\{b\mid (a,b)\in R\}$$
And it is exactly the case that $R=\bigcup_{a\in A}(\{a\}\times R_a)$, almost trivially. You just need to verify that $a\mapsto R_a$ is a well-defined function, but that is just a consequence of it having a specific definition as a set.
It is also quite easy to see that the "sections function" mapping $R$ to $a\mapsto R_a$ is in fact a bijection. Between the relations between $A$ and $B$ and functions from $A$ to $\mathcal P(B)$.