Your question is confusing. A function
maps the members of the set A to members of the set B. The set A is called the domain and B is called the range. (To put it in simple terms anyway.) To define an expression "x - y" is a rational number where x and y are the set of real numbers makes little sense to me.
Now, if you are merely asking what is the size of the set \{ x - y \in \mathbb{Q}|x, y \in \mathbb{R} \} (i.e. The set of all real numbers x and y such that x - y is rational) then your answer is that the size of your set is infinite. Given any value of x we can find an infinite number of y values such that x - y is rational.
I
think this addresses your question. If not, just let me know.
-Dan