The following is a Homework Question that I've been working on and I would like some feedback on my answer: Prove or find a counterexample: For all real numbers $x$ and $y$ it holds that $x + y$ is irrational if, and only if, both $x$ and $y$ are irrational.
So I've got (or at least like to think I've got) a counter example.
Let $x = 1$ and $y = \sqrt{2}$.
Thus $x$ is rational and $y$ is irrational. Adding $x$ and $y$ gives us $1 + \sqrt{2}$ which cannot be simplified any further (right?) and is an irrational number.
Thereby proving that $x + y$ can be irrational without both x and y being irrational.
Is this okay? Any feedback is greatly appreciated thank you!