Suppose $x$ and $y$ are natural numbers. Show that $xy$ odd implies that $x$ and $y$ are both odd.
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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
$endgroup$
add a comment |
$begingroup$
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
$endgroup$
6
$begingroup$
This looks good to me. I especially like that you used the contrapositive rather than deriving an unnecessary contradiction.
$endgroup$
– John Douma
Jan 4 at 8:28
add a comment |
$begingroup$
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
$endgroup$
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
asked Jan 4 at 8:25
user503154
6
$begingroup$
This looks good to me. I especially like that you used the contrapositive rather than deriving an unnecessary contradiction.
$endgroup$
– John Douma
Jan 4 at 8:28
add a comment |
6
$begingroup$
This looks good to me. I especially like that you used the contrapositive rather than deriving an unnecessary contradiction.
$endgroup$
– John Douma
Jan 4 at 8:28
6
6
$begingroup$
This looks good to me. I especially like that you used the contrapositive rather than deriving an unnecessary contradiction.
$endgroup$
– John Douma
Jan 4 at 8:28
$begingroup$
This looks good to me. I especially like that you used the contrapositive rather than deriving an unnecessary contradiction.
$endgroup$
– John Douma
Jan 4 at 8:28
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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.
answered Jan 4 at 8:39
Eevee TrainerEevee Trainer
7,23821337
$endgroup$
3
$begingroup$
Thanks for your feedback! It's really boosting my proof writing confidence :)
$endgroup$
– user503154
Jan 4 at 8:42
add a comment |
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1 Answer
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$begingroup$
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.
answered Jan 4 at 8:39
Eevee TrainerEevee Trainer
7,23821337
$endgroup$
3
$begingroup$
Thanks for your feedback! It's really boosting my proof writing confidence :)
$endgroup$
– user503154
Jan 4 at 8:42
add a comment |
$begingroup$
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.
answered Jan 4 at 8:39
Eevee TrainerEevee Trainer
7,23821337
$endgroup$
3
$begingroup$
Thanks for your feedback! It's really boosting my proof writing confidence :)
$endgroup$
– user503154
Jan 4 at 8:42
add a comment |
$begingroup$
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.
answered Jan 4 at 8:39
Eevee TrainerEevee Trainer
7,23821337
$endgroup$
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.
answered Jan 4 at 8:39
Eevee TrainerEevee Trainer
7,23821337
answered Jan 4 at 8:39
Eevee TrainerEevee Trainer
7,23821337
answered Jan 4 at 8:39
Eevee TrainerEevee Trainer
7,23821337
answered Jan 4 at 8:39
Eevee TrainerEevee Trainer
7,23821337
7,23821337
3
$begingroup$
Thanks for your feedback! It's really boosting my proof writing confidence :)
$endgroup$
– user503154
Jan 4 at 8:42
add a comment |
3
$begingroup$
Thanks for your feedback! It's really boosting my proof writing confidence :)
$endgroup$
– user503154
Jan 4 at 8:42
3
3
$begingroup$
Thanks for your feedback! It's really boosting my proof writing confidence :)
$endgroup$
– user503154
Jan 4 at 8:42
$begingroup$
Thanks for your feedback! It's really boosting my proof writing confidence :)
$endgroup$
– user503154
Jan 4 at 8:42
add a comment |
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6
$begingroup$
This looks good to me. I especially like that you used the contrapositive rather than deriving an unnecessary contradiction.
$endgroup$
– John Douma
Jan 4 at 8:28