Give the equivalence class of 3, -2/3, and 0.
I'm unsure if the equivalence classes for these should just be {3,3}, {2/3,2/3}, and {0,0}. If they extended to the xy=1 condition, I'm unsure about what they would look like in that case. Any help would be appreciated!
Several comments here.
First, it would be good to verify that $ R$ is indeed an equivalence relation.
Second, you note sets like $\{0,0\}$... While this is not incorrect, you should notice that for whatever element $a$, you have $\{a,a\}=\{a\}$.
Third, the equivalence class you mentioned are not correct. For example you have $$\overline{3}=\{3,1/3\}$$
Fourth, the equivalence class of $0$ is a "special case". Do you see why?