On equality of quotient relations

Question

I am a non-mathematician who is taking a self-learning course in mathematics. I am studying a chapter on (equivalence)relations and I have the following question:

Suppose $R$ and $S$ are equivalence relations on a set $A$ and $A/R = A/S$. Is it true that that $R = S$?

Answer 1

Yes: Let $(x,y)\in R$. Then there is a unique $X\in A/R$ such that $x,y\in X$. Since $A/R=A/S$ we have that $X\in A/S$, hence by definition $(x,y)\in S$. The other direction is exactly the same.