Representing / Defining One-to-Many Relationships One can either have an injective, surjective, or bijective function.  What, then, do we do in the case of one-to-many relationships?  If we cannot represent one-to-many relationships as functions are there other mathematical relationships that provide a robust way to explore them?  I understand this is a very popular concept in databases and haven't seen it discussed in math.

One-to-many correspondences can, in fact, be represented as functions. There are several ways of doing this, but I'll explore one possible way.
One important part of the mathematical formalism surrounding mathematical functions is that a function $f$ is a pairing between sets. A function $f$ takes elements of one set $A$, called the domain of $f$, into elements of another set $B$, called the codomain of $f$. For each element $a\in A$ in the domain, we have exactly one element $f(a)\in B$ in the codomain of $f$.
The condition that $f(a)$ be unique is not so restrictive, however, when you consider how general the notion of a set is. All sorts of things can be contained in sets, e.g. numbers, letters, and (here's the kicker) other sets. Perhaps we can define a function into a set of sets, so that for each $a\in A$, we assign a unique $f(a)$ to be a set?
For each set $B$, there is another set $\mathcal{P}(B)$ called the power set of $B$, which is a very specific set of sets. If we choose the codomain of a function to be $\mathcal{P}(B)$, then we can assign to each element $a$ a (unique) subset of $B$, including or disincluding whichever elements of $B$ we want to associate with $a$.
