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Consider the equivalence relation n $\Bbb R$ - {$0$}:

$a$~$b$ if and only if $\dfrac {a}{b} \in \Bbb Q $

In the following list of equivalence classes, find two classes which are equal:

[$\sqrt 3$] , [$1$], [$\sqrt {12}$], [$\sqrt 6$]

I have no idea how to do this question. Please prove the answer and explanations as this question could be on my test.

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  • $\begingroup$ $[x]=[y]$ if and only if $x\sim y$, which is the case if and only if $\frac{x}y$ is rational. Find two numbers in the set $\{\sqrt3,1,\sqrt{12},\sqrt6\}$ whose ratio is rational. $\endgroup$ Dec 13, 2013 at 1:07
  • $\begingroup$ [$\sqrt 3$] , [$\sqrt {12}$]? $\endgroup$
    – user109886
    Dec 13, 2013 at 1:10
  • $\begingroup$ Looks good to me: the ratio is $2$ or $\frac12$, depending on which way you take it, and that’s certainly rational. $\endgroup$ Dec 13, 2013 at 1:16

1 Answer 1

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Think [$\sqrt12$] can be expressed as $2\sqrt3$ right? And you have a [$\sqrt3$]...

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