If $x$ is rational and $xy$ is irrational, then $y$ is irrational.

Question

This is a statement that I need to prove.

Let $x$ and $y$ be real numbers. If $x$ is rational and $x\times y$ is irrational, then $y$ is irrational.

I believe you have to prove this using contradiction. Thanks in advance.

Answer 1

Fill in the blanks proof: suppose $y$ is rational. Then $x\times y$ is $\underline{~~~~~~~~~~~~~~~~}$. Therefore, $y$ cannot be rational, and is hence irrational.

Answer 2

Let's do this as you suggested. By contradiction. Assume that $x$ is rational, $xy$ irrational but $y$ rational. Then, since $x$ is rational, and $y$ is rational, then $xy$ is also rational (why?). But $xy$ is irrational, by assumption. Hence we have a contradiction. (You can reason by contrapositive as well)

Answer 3

Consider contraposition: $(\neg Q \Rightarrow \neg P) \Rightarrow (P \Rightarrow Q)$. If $y$ is rational ($\neg Q$) then $xy$ is rational
($\neg P$) by closure of rationals. Therefore, if $xy$ is irrational ($P$) so is $y$ ($Q$).