Basic question on equivalence relations.

Question

Show that the following relation is an equivalence relation on the given set.

$m \sim n$ in $\mathbb{Z}$ if $m \equiv n\,(\text{mod}\,6)$.

Answer 1

To prove this take into account that

$$ m \equiv n ~ mod ~ 6 \iff \frac{m-n}{6}=0$$

Then:

1. $m-m \equiv 0 ~ mod ~ 6 \implies m \equiv m ~ mod ~ 6$

2. $m-n \equiv 0 ~ mod ~ 6 \implies n-m = 0 ~ mod ~ 6 \implies n \equiv m ~mod ~6$

3. If $m-n \equiv 0 ~ mod ~ 6$ and $n-p \equiv 0 ~ mod ~ 6$ then $(m-n) + (n-p) = m - p \equiv 0 ~ mod ~ 6 \implies m \equiv p ~ mod ~ 6$