Equivalence relation question help with x+y

Question

Need help with this question:

Explain whether the relation $(x\sim y)=\left\{(x,y)∈N\times N\bigg|x+y\text{ is even}\right\}$ is transitive where N is the set of Natural numbers.

I've tried to work this question out but the $+$ sign is confusing me, if it was a $<$ (less than) then I would be able to solve it but I don't know what it means in equivalence relation when its $x+y$. Someone please help.

Answer 1

The relation isn't $x+y$. The relation is "$x\sim y$ iff $x+y$ is even" or equivalently, $x\sim y$ iff $x$ and $y$ have the same parity.

The rephrasing should make the transitivity of the relation immediate.

Answer 2

$2\mid(x+y) \land 2\mid(y+z) \implies 2\mid((x+y)-2y+(y+z)) = x+z$ because obviously $2\mid 2y$