Suppose $R$ be the relation on $[0,1)$ s.t. $aRb\iff a-b$ is rational. Let $[0]$ be the equivalence class with respect to relation $R$ on $[0,1)$. Let $Q$ be the set of all equivalence class on [0,1). Prove that for any $W\in Q$, there exist a bijection $f:[0]\to W$, where $f(z)=z+g$ if $z+g<1$ and $f(z)=z+g-1$ when $z+g\ge1, g\in [0,1)$ and $z\in [0]$.
Attempts: I know that $[0]$ is indeed the set of all rational number in $[0,1)$. So when $W=Q$, it is trivial by definining the function letting $f(z)=z,g=0$. When $A\ne Q$, then $A$ is set of irrational numbers $\in [0,1)$ but then i am not so sure how to prove it since there are uncountably many irrational numbers