Q:

why the Vitali set has size $2^{\aleph_0}$?

why the Vitali set has size $2^{\aleph_0}$ and therefore there is a bijection between $\mathbb R$ and the Vitali set? thanks

A:

Note that there are $2^{\aleph_0}$ equivalence classes in $\Bbb{R/Q}$. Since a choice function from these classes is injective, this shows that there are exactly $2^{\aleph_0}$ elements in a Vitali set, and certainly there are not more.
To see that there are $2^{\aleph_0}$ equivalence classes, note that the vector spaces $\Bbb R$ and $\Bbb{R/Q}$ are isomorphic as vector spaces over $\Bbb Q$. Therefore they have the same dimension and cardinality.

