A question about rational and irrational

Question

The original statement is: "Suppose x and y are irrational. Prove or disprove the following statement: Either x + y or xy is irrational."

So I use contraposition to prove it which is:

If x + y and xy are rational, then x or y is rational.

But then the sum of a rational number and an irrational number is irrational which would make my statement false. What did I do wrong here?

Answer 1

You forgot the 'Prove or disprove' part of the question. The implication is false so you have to give a counter-example. Take $x=\sqrt 2$ and $y =-\sqrt 2$.

Answer 2

What is wrong is that the counterpositive is:

If $x+y$ and $xy$ are both rational or both irrational, then $x$ or $y$ is rational.