What is the uniqueness of the equivalence relation? In Lee, 'Introduction to Topological Manifolds', Appendix A, Exercise A.2 I am asked to prove that if $\mathcal C$ is a partition of $X$ there is a unique equivalence relation $\sim$ such that classes of equivalence of $\sim$ are elements of $\mathcal C$. Since the definition of difference/equality of equivalence relations was not given, I thought that it should be based on the equivalence classes of such a relation.
More formal, am I right stating that 

There is a unique equivalence relation $\sim$ which admits sentence $S$ iff for any $\sim^1$ and $\sim^2$ admitting $S$ their classes of equivalence are the same?


I'd define equality of equivalence relations, or any other relations, as follows.  A relation $\sim_1$ is equal to a relation $\sim_2$ precisely if
$$
\forall a\  \forall b\  (a\sim_1 b \text{ if and only if }a\sim_2 b).
$$
And that would also apply to other relations than equivalence relations.
