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Let x,y be real numbers. Define the relation S as

x S y if |x - y| $\epsilon$ Q

where Q is the set of rational numbers.

Find all equivalence classes of S.


I work in the undergrad tutor center and have gotten the above question both this year and last. I believe it is problem 13.14.4b in the book "How to Prove it: A Structured Approach , Daniel J. Velleman, 2nd Ed", but I don't own the book myself to check.

I don't see how it's possible to list all of the equivalence classes of S. For instance, if we ask the simpler question of whether the number $\pi + e$ is related to $0$, that is asking whether integers $p$ and $q$ exist such that

$\pi + e = \frac{p}{q}$

Which is currently an open problem. So if we cannot even establish whether $\pi + e$ and $0$ are in the same equivalance class, how can we hope to find them all?

Am I overthinking it? Is this question solvable?

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  • $\begingroup$ A solution to this problem is equivalent to a basis for the vector space $\mathbb{R}$ over $\mathbb{Q}$, which has uncountable dimensionality. Not sure if that matters. $\endgroup$ Nov 19, 2014 at 22:50

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You can’t list them, but you can describe them, in the sense that for each $x\in\Bbb R$ you can write down a simple description of the equivalence class of $x$: it’s $x+\Bbb Q=\{x+q:q\in\Bbb Q\}$; in all likelihood this is the sort of answer that was intended, though one could wish that the question were worded better.

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