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In $\mathbb{R}$ let $a\sim b$ iff $a-b$ exists $\in$ $\mathbb{Q}$.

I have already proven that $\sim$ is an equivalence relation.

However, the second part of the question asks to describe the partition associated with the equivalence relation, and this is where I am confused.

How would you describe the partition with this equivalence relation?

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A tricky answer :

$$P = \bigcup_{x\in \Bbb R} \{ \{x+y | y \in \Bbb Q \} \}$$

The "problem" is that in this union, you have multiple time the same element in this description

Now, if you want an union where each element appear only once in its description, it's more complicated and I'm 99% certain that you'll need the axiom of choice/Zorn lemma.

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There are infinitely many sets in the partition; in fact, uncountably many. Each set in the partition is a copy of $\mathbb{Q}$. The union of these copies of $\mathbb{Q}$ make up the whole real line $\mathbb{R}$.

If you want to get a set of representatives—picking one real number from each set in the partition—it will look pretty weird. You can find a full set of representatives in an arbitrarily small interval, consisting of an unmeasurable collection of points. Such a set of representatives is called a Vitali set.

See also Visualizing quotient groups: $\mathbb{R/Q}$.

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