# Can this recursive definition be improved?

The question goes something like this:
Give a recursive definition for the following set $S$: The set of ordered pairs $(a,b)$ where $a$ and $b$ are both positive integers and $a + b$ is odd.

In order to solve this, I first had to create a recursive case for odd and even numbers as follows.

$Q(1) = 1 || Q(N) = (Q(n-1)) + 2, N > 1$ //this gives all our positive odd values
$R(2) = 2 || R(N) = (R(n-1)) + 2, N > 2$ // this gives all our positive even values
If $x$ is in set $Q$, and $y$ is in set $R$, then $(x,y)$ is in $S$ and $(y,x)$ is in $S$

Would this be correct, and is there a different way I Should be doing this? I see that this question was asked here, however their approach was different than mine and I'm not entirely sure that I understand how they are doing it.