Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This is a question from my textbook, and I'm pretty sure I have the answer, but I am having trouble writing out what I am thinking.

Prove: If $n$ is an integer and $3n+2$ is odd, then $n$ is odd.

Contrapositive: If $n$ is even, then $3n+2$ is even.

$n$ is even, by definition $n = 2k$, where $k$ is an integer.

Plugging in $2k$ into $3n+2$, we have $3(2k)+2$.

$3(2k)+2 = 6k+2 = 2(3k+1)$, which means $3n+2$ is even.

Since if $n$ is even, then $3n+2$ is even. Negating the conclusion of the original conditional statement implies that it's hypothesis "$3n+2$ is odd" is false, therefor if $3n+2$ is odd, $n$ is odd.

What am I missing, or what can I do to make this less awkward? Or am I wrong about the proof on the most basic levels?

share|improve this question
    
It can be proved directly: $\rm\ 3n\!+\!2\,=\, 2k\!+\!1\,\ is\ odd\:\Rightarrow\: n\,=\,2(k\!−\!n)-1\,\ is\ odd.$ –  Math Gems Feb 26 '13 at 6:03

1 Answer 1

up vote 2 down vote accepted

Your proof is correct, though you could improve on it by writing it in essay style, cutting down on some verbosity, and by mentioning that $3k+1$ is also an integer. For example, you could write something like this:

We try to prove the contrapositive. If $n$ is even, then $n=2k$ for some integer $k$, so $3n+2=3(2k)+2=6k+2=2(3k+1)$ where $3k+1$ is an integer, making $3n+2$ even.

Note that proofs can be written with varying degrees of detail. In beginning courses, instructors prefer to see more detailed proofs to ensure students really understand what they are writing.

share|improve this answer
    
Up to how much? Do I remove everything after "Since if n is even, then 3n+2 is even."? –  Chase Feb 24 '13 at 1:47
    
So it's better to write out explicitly what I have up there? (As in, "3(2k)+2=6k+2=2(3k+1), which means 3n+2 is even." should have been a bit more spelled out) –  Chase Feb 24 '13 at 1:55
    
I see. I have trouble trying with proofs and explaining proofs so thank you! –  Chase Feb 24 '13 at 2:02

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.