What is the range of "x-y is a rational number"? Where x and y are set of real numbers.
is it -- (0,0), (1/2,-1/2), (1/3, 4/3)?
Your question is confusing. A function $\displaystyle f:A \to B$ 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 $\displaystyle \{ x - y \in \mathbb{Q}|x, y \in \mathbb{R} \}$ (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
You have a relation $\displaystyle R$ such that $\displaystyle \left( {x,y} \right) \in R \Leftrightarrow \,\left( {x - y} \right) \in Q$.
The range of any relation is the set of second terms.
What is the set of all second terms in $\displaystyle R$? How is a difference of two reals rational?
Here are examples.
$\displaystyle \left( {2,\frac{1}{2}} \right) \in R$ because $\displaystyle \left( {2 - \frac{1}{2}} \right)$ is rational.
$\displaystyle \left( {\pi ,\pi } \right) \in R$ because $\displaystyle \pi - \pi = 0$ and zero is rational.
Does that mean that $\displaystyle R$ is reflexive?