Prove if x is even, then x+1 is odd.

Let x be an integer and prove if x is even, then x+1 is odd.

We will use contraposition for this proof, so we start by assuming that x + 1 is not odd.

Then we can say there exists an integer k such that x + 1 = 2k

so x = 2k - 1 = 2(k-1) + 1

Thus x = 2(k-1) + 1 which is odd. So we can no say if x + 1 is not odd, then x is not even. Thus, by contraposition, if x is even, x+1 is odd.