Equivalence relation $a\sim b$ $ \iff a-b\in\mathbb{Z}$

Question

I have this equivalence relation defined on $\mathbb{Q}$ and $a\sim b$
$ \iff a-b\in\mathbb{Z}$

I know this is an equivalence relation and have proven so already. But how can I prove that for rational numbers $a,b,c$ we have $a\sim b$ $ \iff a+c\sim b+c$?

I was wondering how I could go about proving this? I was thinking to combine the relations since they are obviously related to each other so $a+c\sim b+c \iff a-b\in\mathbb{Z}$, then would I try and prove the 3 criteria? And if so how? Thank you!

Answer 1

$a\sim b\iff a-b\in\mathbb Z \iff (a+c)-(b+c) \in \mathbb Z \iff a+c\sim b+c$

Answer 2

For any $c\in\Bbb Q$, we can reason that $$a\sim b\iff\exists k\in\Bbb Z(a-b=k)\iff\exists k\in\Bbb Z((a+c)-(b+c)=k)\iff a+c\sim b+c$$ because $(a+c)-(b+c)=a-b$.