Consider an equivalence relation θ on some set B and a function:
$:^2→ f : B 2 → B $.
We want to lift f to the set of equivalence classes B/θ, i.e., we want to define a function $:(/θ)^2→(/θ) $ canonically in terms of $f$. For this to be meaningful, f has to be θ-consistent. That is, if $f$ is applied to a pair $(_1,_2)∈^2$, the equivalence class $[(_1,_2)]_θ$ may only depend on the equivalence classes $[_1]_θand [_2]_θ $(irrespective of which concrete elements both b_1 and b_2 are within their equivalence classes).
Task: Define the sum of two fractional representations of rational numbers as a function sum: $^2 → A$ (for A = Z × (Z \ {0}) as defined above), using standard addition and multiplication in the integers
I defined a function $+:^2→:[(,)]+[(,)]⇔[(+,)]$
So taking two equivalence classes, this functions provides, unimportant of the rational representation chosen, the sum of two rationals. Could someone help me with this problem as I don't know whether my solution is correct?