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.

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    $\begingroup$ Unless you are using this function somehow to construct the negative real numbers or to define subtraction what you suggest is perfectly OK. You can edit the question to include more information if you're still unsure. $\endgroup$ – Ethan Bolker Dec 7 '18 at 21:13
  • $\begingroup$ Equivalence classes under what equivalence relation? $\endgroup$ – user4894 Dec 7 '18 at 21:19

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 $\bigcup_i S_i = S$.


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