I am given the following quotient set:
$$A/\sim_R = \{ \{ 1,7 \}, \{ 3 \}, \{ 4, 5, 9 \} \}$$
For which I need to trace back the original set $A$ and relation of equivalence $R$.
Up until now, I have figured out how to find equivalence classes for each element, given the set $A$ and the relation $R$. However, this sounds like doing it backwards.
My main question is: Is there any way to reconstruct the relation $R$? I know how to deduce the elements $A$, since the quotient set is a partition of the set $A$ and the union of all elements of the quotient set gives me the set $A$. However, let's take for example $ \{ 1, 7 \}$ as a single element of the quotient set. Do I just take $(1,7), (7,1), (1,1), (7,7)$ as members of the relation $R$, and do so for the rest of the elements of the quotient set?