Proof By ContraPositive: if xy is odd, then x is odd and y is odd

Question

If $xy$ is odd, then $x$ is odd and $y$ is odd

I was just wondering if the correct contrapositive would involve proving these three cases:

1) $x$ is even or $y$ is odd

2) $x$ is odd or $y$ is even

3) $x$ is even or $y$ is even

I'm not too sure about if the last case is necessary as in the answer to this question only the first two cases were shown. I guess what i'm trying to ask is why do we not check the last case, since it is a possible negation of $x$ is odd and $y$ is odd?

Answer 1

The counter-positive of the assertion

if $xy$ is odd, then $x$ and $y$ are odd

is

if $x$ is even or $y$ is even, then $xy$ is even.

And asserting that $x$ is even or $y$ is even is equivalent to asserting that we have one of the following possibilities:

1. $x$ is even and $y$ is odd;
2. $x$ is odd and $y$ is even;
3. both $x$ and $y$ are even.