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I need to prove that every equivalence class created by the equivalnce relation $\sim$ on $\mathbb{R}$, that is defined by: $a\sim b \Leftrightarrow (a-b) \in \mathbb{Q}$, is $\aleph_0$. Furthermore, I need to prove that the group of all equivalence classes is uncountable. I have an idea how to move from step one (proving it's $\aleph_0$) to step two (proving that the group of all equivalence classes is uncountable), by using the laws of addition in cardinality. However, I have no idea how to even approach the first step. Any help or idea will be appreciated!

Thanks a bunch!

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    $\begingroup$ Let $B$ be any equivalence class, and let $b$ be a fixed element of $B$. Then $B$ consists of all $a$ of the form $b+r$, where $r$ ranges over the rationals. This gives us a one to one correspondence between $B$ and the rationals. $\endgroup$ – André Nicolas Jan 31 '14 at 0:03
  • $\begingroup$ Related. $\endgroup$ – Cameron Buie Jan 31 '14 at 19:21
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Hint: Take any $b\in\Bbb R$. Then $a\sim b$ if and only if $a=b+p$ for some $p$ such that...what?

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  • $\begingroup$ Can't believe I failed to see that. Thanks a lot!! $\endgroup$ – Studentmath Jan 31 '14 at 0:09
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HINT: The rationals are countable. If $x-y\in\Bbb Q$ then there is a unique $q\in\Bbb Q$ such that $x+q=y$.

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