# Thread: Proofs can't figure it out

1. ## Proofs can't figure it out

I am completely lost on how to do this.
Prove that an integer n is even if and only if n+1 is odd

thanks in advance for any help

2. For starters make n+1 odd as follows

2(n+1)+1 now show the previous number is even.

3. ## even odd proof

T.P. n is even iff n+1 is odd

Assume n is even then let n = 2k for some k element of Z (definition of even)

Then n+1 = 2k + 1, but 2k + 1 is the definition of an odd number. This proves the forward direction now reverse it.

Let n + 1 be odd then by definition of an odd number n + 1 = 2k + 1 for an arbitrary k element of Z

Now subtract 1 from both sides and you have n = 2k which by definition is even.

This completes the proof in both directions.