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Can anyone confirm if my proof is correct, please?

Claim:- “If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.”

Proof:-

Assume that both $x$ and $y$ are rational. Now, let $x = \dfrac pq$ and $y = \dfrac mn$ since both of them are rational. $xy =\dfrac pq * \dfrac mn = \dfrac{pm}{qn}$ Thus, if the product $xy$ can be written as a fraction, it's not a irrational number. Therefore if one of $x$ and $y$ is not irrational, then the product is not irrational.

By the principle of proof by contraposition, If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.

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    $\begingroup$ Please do not deface your questions. $\endgroup$ – user642796 Oct 1 '13 at 13:18
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I would specify that $p,q,m,n\in\mathbb{Z},q\neq0,n\neq0$, in your first line of the proof.

Other than that, the rest looks good.

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Looks good to me. ${}{}{}{}{}{}{}{}$

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