# Describing an equivalence class given a contained number.

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?

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 $\sim{2\over 3}$?
• 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. – JayVB Nov 18 '17 at 0:13
• @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 $\sim{2\over 3}$? – Noah Schweber Nov 18 '17 at 0:14
• I am looking for values such that $\frac{2}{3} - x_2$ is equal to an integer? This isn't my strongest topic. – JayVB Nov 18 '17 at 0:30
• My main issue is understanding how $[\frac{2}{3}]$ relates to the question. – JayVB Nov 18 '17 at 0:37
• @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. – Noah Schweber Nov 18 '17 at 1:19