Prove the following statement by proving its contrapositive: if $r$ is irrational, then $r^\frac{1}{5}$ is irrational.
Its contrapositive will be:
If $r^\frac{1}{5}$ is not irrational, then $r$ is not irrational.
How can I prove the contrapositive ?
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1Well, first of all you can replace "not irrational" by "rational". Now see if you can take it from there. – TonyK Aug 10 '14 at 11:31
2 Answers
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If $r^{1/5}$ is rational, then there exist $p,q \in \mathbb{Z}$ with $q\neq 0$ so that $r^{1/5}=\frac{p}{q}$. Therefore $r = \frac{p^5}{q^5}$ is a rational number since $\mathbb{Q}$ is closed under product.
Darth Geek
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2"since $\mathbb Q$ is closed under product": simpler just to point out that $p^5$ and $q^5$are integers if $p$ and $q$ are. – TonyK Aug 10 '14 at 14:48
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Suppose $r^{1/5}$ rational. Then $r^{1/5}=\frac{p}{q}$ for certain $p,q\in\mathbb Q$. We can suppose $\gcd(p,q)=1$. Then $r=(r^{1/5})^5=\frac{p^5}{q^5}\in\mathbb Q$
Q.E.D.
idm
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You are combining reductio ad absurdum with contrapositive. But only contrapositive is aked. The reductio ad absurdum is unnecessary. – Darth Geek Aug 10 '14 at 11:41
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ok sorry, i'm not english, I didn't understand the term, but it's fine, I corrected (but the two proof are actually the same) – idm Aug 10 '14 at 11:43
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Your proof is one sentence too long: gcd$(p,q)$ doesn't have to be $1$ here. – TonyK Aug 10 '14 at 14:49