I was asked to prove that the difference of two odd numbers is even but, given that I might be the most distracted person in the world, I proved the sum. However, I think it should still be correct but I'll first show you my proof.
Let $n$ and $m$ be odd integers. Then $\exists k,l\in\mathbb Z:m=2k+1\ \land n=2l+1$ . Then, $$m+n=(2k+1)+(2l+1)$$ $$=\ 2k+2l+2$$ $$=2(k+l+1)$$ Since $k+l+1$ is an integer, then $2(k+l+1)$ is even.
I told my professor that this proof still worked for the difference of odd numbers because if you make $n$ a negative integer, then that is basically subtraction. He says that that would've been correct had I specified that $n$ was negative. But I find that redundant because by definition an integer can be either positive or negative. Then this proof should work for the four cases in which a) both $m$ and $n$ are positive, b) $m$ is positive and $n$ is negative, c) $m$ is negative and $n$ is positive, and d) both $m$ and $n$ are negative.
I didn't feel it was right to with him, given he's a professor and I'm just an undergrad, but I just can't really understand the difference. I hope you can help me understand and correct me if I'm wrong.