Uncountability of the equivalence classes of $\mathbb{R}/\mathbb{Q}$ Let $a,b\in[0,1]$ and define the equivalence relation $\sim$ by $a\sim b\iff a-b\in\mathbb{Q}$. This relation partitions $[0,1]$ into equivalence classes where every class consists of a set of numbers which are equivalent under $\sim$,
My textbook states (without proof):

The set $[0,1]/\sim$ consists of uncountably many of these classes, where
  each class consists of countably many members.

How can I formally prove this statement?

If you know $\Bbb Q$ is countable, that covers the second half.  Then use the fact that a countable union of countable sets is again countable to show that there must be uncountably many classes.
