Hi

Here's a problem I am trying to solve

Find the equivalence classes of this relation

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

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

so

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

(is that correct ?) So

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 , the base of natural logarithm is an irrational number ,

so the equivalence classes are and set corresponding to

each irrational number.

is my reasoning correct ?