The mathematical idea of functions is the set of ordered
pairs, where the left item in the pair is a member of the
domain and the right item is a member of the range.
The domain can then be considered also to be ordered
pairs, so that + then looks like
{((1,1),2), ((2,2),4) ... }
And so on. This is a consequence of the foundational
approach in which sets are the basic building blocks.
(And ordered pairs are defined in terms of sets, via
Kuratowski or whatever.)
However there is another way to go, which I first
encountered in ch. 4 of TTM, and which I don't see
discussed much or at all in mathematical circles,
and which I find immensely appealing. The approach
is to work directly with relations. So + looks like
{(x=1, y=1, z=2), (x=2, y=2, z=4) ...}
In point of fact, there is almost nothing of significance
to this idea; it's equivalent to the usual formulation.
But I nonetheless find the idea appealing of thinking
about the foundations of mathematics in terms of
taking relations as primitive rather than sets.
