Hi

Here's a problem I am trying to solve

Find the equivalence classes of this relation

$\displaystyle S=\{(x,y)\in \mathbb{R}\times \mathbb{R} \vert \;x-y \in \mathbb{Q}\}$

now this is an equivalence relation. If we choose any rational number x , then we can see that

$\displaystyle [x]_{S}=\mathbb{Q}$

so Q itself is an equivalence class of S. Now consider irrational number like pi

$\displaystyle \pi \in \mathbb{R}$

$\displaystyle \pi-\pi = 0 \in \mathbb{Q}$

so $\displaystyle (\pi,\pi)\in S$

$\displaystyle \therefore \pi \in [\pi]_{S}$

if we subtract any other irrational or rational number from pi , it will not be in

$\displaystyle \mathbb{Q}$ (is that correct ?) So

$\displaystyle [\pi]_{S} =\{\pi\}$

similar conclusions can be drawn for any other irrational number. so other equivalence classes would correspond to irrational numbers where the set will

contain only that irrational number. for example , since $\displaystyle \mathit{e}$ , the base of natural logarithm is an irrational number ,

$\displaystyle [\mathit{e}]_{S}=\{\mathit{e}\}$

so the equivalence classes are $\displaystyle \mathbb{Q}$ and set corresponding to

each irrational number.

$\displaystyle \blacksqaure$

is my reasoning correct ?