Why is this function well definded?

Question

I have to take a look at the relation of $ x \sim y: \Leftrightarrow \exists k \in \mathbb{Z} : x-y=5k$ in $\mathbb{Z}$. I have no idea how to show at $\mathbb{Z} / \sim$ that the addition is well defined by $ \bar{x} +\bar{y} := \overline{x+y} $. Is there someone who can help me, please?

Answer 1

HINT:Denote by $[x], [y]$ the equivalence classes. To test that the addition is well defined you must see that $[x+y]=[x'+y']$, for all $x'\in[x],y'\in[y]$. In other words: $$x+y\sim x'+y'.$$