Suppose $x$ and $y$ are natural numbers. Show that $xy$ odd implies that $x$ and $y$ are both odd.












2














Is my following proof correct using the contrapositive method?



Contrapositive Statement:



Suppose $x$ and $y$ are natural numbers. Show that $x$ or $y$ is even implies that $xy$ is even.



Proof:



For $x,yin mathbb{N}$, assume, without loss of generality, that $x$ is even. Then $x=2m$ for some $min mathbb{N}$. Therefore,$$xy=(2m)y=2r,$$ where $r=myin mathbb{N}$. Thus, $xy$ is even.






proof-verification proof-writing







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  • 6




    This looks good to me. I especially like that you used the contrapositive rather than deriving an unnecessary contradiction.
    – John Douma
    Jan 4 at 8:28


















2














Is my following proof correct using the contrapositive method?



Contrapositive Statement:



Suppose $x$ and $y$ are natural numbers. Show that $x$ or $y$ is even implies that $xy$ is even.



Proof:



For $x,yin mathbb{N}$, assume, without loss of generality, that $x$ is even. Then $x=2m$ for some $min mathbb{N}$. Therefore,$$xy=(2m)y=2r,$$ where $r=myin mathbb{N}$. Thus, $xy$ is even.






proof-verification proof-writing







share|cite|improve this question


















  • 6




    This looks good to me. I especially like that you used the contrapositive rather than deriving an unnecessary contradiction.
    – John Douma
    Jan 4 at 8:28
















2












2








2







Is my following proof correct using the contrapositive method?



Contrapositive Statement:



Suppose $x$ and $y$ are natural numbers. Show that $x$ or $y$ is even implies that $xy$ is even.



Proof:



For $x,yin mathbb{N}$, assume, without loss of generality, that $x$ is even. Then $x=2m$ for some $min mathbb{N}$. Therefore,$$xy=(2m)y=2r,$$ where $r=myin mathbb{N}$. Thus, $xy$ is even.






proof-verification proof-writing







share|cite|improve this question













Is my following proof correct using the contrapositive method?



Contrapositive Statement:



Suppose $x$ and $y$ are natural numbers. Show that $x$ or $y$ is even implies that $xy$ is even.



Proof:



For $x,yin mathbb{N}$, assume, without loss of generality, that $x$ is even. Then $x=2m$ for some $min mathbb{N}$. Therefore,$$xy=(2m)y=2r,$$ where $r=myin mathbb{N}$. Thus, $xy$ is even.






proof-verification proof-writing




proof-verification proof-writing






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share|cite|improve this question











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asked Jan 4 at 8:25









ImranImran

1815




1815








  • 6




    This looks good to me. I especially like that you used the contrapositive rather than deriving an unnecessary contradiction.
    – John Douma
    Jan 4 at 8:28
















  • 6




    This looks good to me. I especially like that you used the contrapositive rather than deriving an unnecessary contradiction.
    – John Douma
    Jan 4 at 8:28










6




6




This looks good to me. I especially like that you used the contrapositive rather than deriving an unnecessary contradiction.
– John Douma
Jan 4 at 8:28






This looks good to me. I especially like that you used the contrapositive rather than deriving an unnecessary contradiction.
– John Douma
Jan 4 at 8:28












1 Answer
1






active

oldest

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1














Your proof is fine. I like the insight of choosing to use the contrapositive - that's a skill that will serve you well, because it makes the proof swifter and more elegant here, in my personal opinion.



It might be worth spelling out in your proof that this in the contrapositive, and thus implies the result you intended to prove, just for completeness' sake, though. A minor nitpick but nothing huge.






share|cite|improve this answer

















  • 1




    Thanks for your feedback! It's really boosting my proof writing confidence :)
    – Imran
    Jan 4 at 8:42











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1 Answer
1






active

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active

oldest

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active

oldest

votes









1














Your proof is fine. I like the insight of choosing to use the contrapositive - that's a skill that will serve you well, because it makes the proof swifter and more elegant here, in my personal opinion.



It might be worth spelling out in your proof that this in the contrapositive, and thus implies the result you intended to prove, just for completeness' sake, though. A minor nitpick but nothing huge.






share|cite|improve this answer

















  • 1




    Thanks for your feedback! It's really boosting my proof writing confidence :)
    – Imran
    Jan 4 at 8:42
















1














Your proof is fine. I like the insight of choosing to use the contrapositive - that's a skill that will serve you well, because it makes the proof swifter and more elegant here, in my personal opinion.



It might be worth spelling out in your proof that this in the contrapositive, and thus implies the result you intended to prove, just for completeness' sake, though. A minor nitpick but nothing huge.






share|cite|improve this answer

















  • 1




    Thanks for your feedback! It's really boosting my proof writing confidence :)
    – Imran
    Jan 4 at 8:42














1












1








1






Your proof is fine. I like the insight of choosing to use the contrapositive - that's a skill that will serve you well, because it makes the proof swifter and more elegant here, in my personal opinion.



It might be worth spelling out in your proof that this in the contrapositive, and thus implies the result you intended to prove, just for completeness' sake, though. A minor nitpick but nothing huge.






share|cite|improve this answer












Your proof is fine. I like the insight of choosing to use the contrapositive - that's a skill that will serve you well, because it makes the proof swifter and more elegant here, in my personal opinion.



It might be worth spelling out in your proof that this in the contrapositive, and thus implies the result you intended to prove, just for completeness' sake, though. A minor nitpick but nothing huge.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 4 at 8:39









Eevee TrainerEevee Trainer

5,0211734




5,0211734








  • 1




    Thanks for your feedback! It's really boosting my proof writing confidence :)
    – Imran
    Jan 4 at 8:42














  • 1




    Thanks for your feedback! It's really boosting my proof writing confidence :)
    – Imran
    Jan 4 at 8:42








1




1




Thanks for your feedback! It's really boosting my proof writing confidence :)
– Imran
Jan 4 at 8:42




Thanks for your feedback! It's really boosting my proof writing confidence :)
– Imran
Jan 4 at 8:42


















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