Find equivalence classes of $\sim$ on $\mathbb{Z}$

Question

Find equivalence classes of $\sim$ on $\mathbb{Z},\;\; m \sim n \iff |m-3|=|n-3|$.

I tried to write absolute values as: $n=6-m, \; n = m.\;$ For, $m=0$ we have $n=6$ so we can say $0 \sim 6$, but I cannot make it. Thanks in advance!

Answer 1

The equality $|m-3|=|n-3|$ means that one of the following possibilities takes place:

• $m-3=n-3$;
• $m-3=-(n-3)$.

In the first case, you have $n=m$, whereas in the second case you have $n=6-m$.