I would suggest you go back and reread the problem. is never odd.
Your title, however, says , not . Perhaps if you tried proving that you will do better.
Let n be an element of Z. Prove that is odd if and only if is even.
Since it's a biconditional, I'm certain there will be two parts: 1) Proving , and 2) proving
It's in the proof by contrapositive section, so I'll try to prove it that way.
1) If is odd, then is even.
Contrapositive: If is odd, then is even.
Assuming is odd, then
for some integer m.
Therefore, is even and the implication is true.
My professor says that my proof is wrong, but I don't see how.
I would suggest you go back and reread the problem. is never odd.
Your title, however, says , not . Perhaps if you tried proving that you will do better.
Correction: Let n be an element of Z. Prove that is odd if and only if is even.
1) If is odd, then is even.
Contrapositive: If is odd, then is even.
Assuming is odd, then
for some integer m.
Therefore, is even and the implication is true.