Non-measurable sets construction

Question

In explanation of non-measurable sets we take $Q_1=Q \cap [-1,1]$ (Q is set of rational numbers). Then we define equivalence relation on $[0,1]$ $x,y \in [0,1]$ as $x\sim y$ iff $x-y \in Q_1$. This equivalence relation introduces a partition on [0,1] splitting [0,1] into mutually disjoint classes $E_\alpha$. Then [0,1] = $\cup E_\alpha$. Then it is said each element of $E_\alpha$ is rational. How can each element of $E_\alpha$ be rational. What will happen to irrational numbers of [0,1].