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I have a question where I am asked to decide whether $∼$ is an equivalence relation on $\Bbb R$ where $x_1∼x_2$ means $x_1-x_2$ is an integer. I have managed to prove it is Reflexive, Symmetric and Transitive but later I am asked to "Describe $[\frac{2}{3}]$, the equivalence class containing $\frac{2}{3}$" if the relation is an equivalence relation (Which is it).I have never come across this phrase, could someone explain this to me?

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It's just asking you to say what the set of elements equivalent to ${2\over 3}$ is.

For example, if we looked at the equivalence relation $\approx$ on $\mathbb{Z}$ given by $a\approx b$ iff $a^2=b^2$, we'd have $[2]=\{-2,2\}$. So:

What numbers are $\sim{2\over 3}$?

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  • $\begingroup$ I might be completely wrong here but would it be $\frac{2n}{3n}$, I completely understand your example but my one I'm kinda confused. $\endgroup$ – JayVB Nov 18 '17 at 0:13
  • $\begingroup$ @JayVB No - $2n\over 3n$ is just $2\over 3$ (unless $n=0$ in which case it's undefined). Start small: what's another number which is $\sim{2\over 3}$? $\endgroup$ – Noah Schweber Nov 18 '17 at 0:14
  • $\begingroup$ I am looking for values such that $\frac{2}{3} - x_2$ is equal to an integer? This isn't my strongest topic. $\endgroup$ – JayVB Nov 18 '17 at 0:30
  • $\begingroup$ My main issue is understanding how $[\frac{2}{3}]$ relates to the question. $\endgroup$ – JayVB Nov 18 '17 at 0:37
  • $\begingroup$ @JayVB $[x]$ denotes the equivalence class of $x$. So yes, you are being asked to describe all the values which, when subtracted from $2\over 3$, yield an integer. $\endgroup$ – Noah Schweber Nov 18 '17 at 1:19

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