When defining equivalence classes using elements contained by the nonnegative reals, may I use subtraction in the function that defines equivalence between those classes? My thinking is that if subtraction is defined for the reals it could be used but if my elements are strictly nonnegative, I'm not sure if that makes a difference.
The equivalence classes on a set $S$ can be seen a partition of $S$ in (disjoint) subsets $S_i$. You can define those subsets, i.e. each equivalence class, in any way you wish. As long as you define them to form a partition, that is such the sets $S_i$ are disjoint subsets of $S$ and satisfying $\bigcup_i S_i = S$.