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?


By definition, an equivalence relation on a set $S$ is a subset of $S\times S$ fulfilling some conditions. This gives you a natural notion of equality for equivalence relations: Two equivalence relations on $S$ are equal if they are given by the same subset of $S\times S$.
It is a general fact that an equivalence relation is uniquely determined by its equivalence classes. This is -- somewhat tautologically -- because two elements are equivalent if and only if they belong to the same equivalence class.
