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Let S be a set and $\{S_\alpha\}$ be nonempty subsets such that S = $\bigcup_{\alpha} $ $S_\alpha$ and $S_\alpha \cap S_\beta$ =$\emptyset $ if $\alpha \neq \beta $ Define an equivalence relation on S in such a way that the $S_\alpha$ are precisely all the equivalence classes.

I don't understand what this is asking. What am I supposed to do?

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    @Sigur I would say that if 2 classes aren't disjoint, then they must be equal. But how do I put that into math/equivalence-y notation?2012-10-16

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What about this: given $x,y\in S$, they are equivalent if and only if there exists $\alpha$ such that $x,y\in S_\alpha$. Now you can prove that this is an equivalence relations. Note that it's fundamental to have the union equal to $S$.

  1. $x\sim x$, for all $x\in S$;
  2. if $x\sim y$ then $y\sim x$, for all $x,y\in S$;
  3. if $x\sim y$ and $y\sim z$ then $x\sim z$, for all $x,y,z\in S$.
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    Yeah I get it now, thank you very much!2012-10-16