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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?

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See this question. – Zev Chonoles Oct 16 '11 at 16:35
@ZevChonoles: thank you, I was able to prove this fact - I just wonder what did he mean with the word unique. Though, if the relation is a set, then I know have to differ relations. Just would like to know if there is other definition of uniqueness – Ilya Oct 16 '11 at 16:38
up vote 3 down vote accepted

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.

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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.

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