Looking for tips to prove the homework
$\forall x,y \in \mathbb{R}, x \in \mathbb{Q} \land y \notin \mathbb{Q} \implies (x + y) \notin \mathbb{Q}$
Can I assume the hypothesis and to yield a contradiction assume that $(x + y)$ is rational, or rather, how should I approach it? So far I know the definitions for an integer $n$ being odd, even, divisible by an integer $m$, has a remainder $r$, and the definition of a rational having integers $p, q$ s.t. $q \neq 0$ and $\frac{p}{q} = Q$
Is there some other tool I might find useful for solving this particular problem? How might I use it? Thanks.
An approach by contradiction is a good idea. Suppose $x+y\in\mathbb{Q}$. What happens when you subtract $x$? Recall or prove that $\mathbb{Q}$ is closed under addition (and hence subtraction).