The equivalence relation $\sim$ on the set of transcendental $X$ is defined by the following property:
$x\sim y$ if and only if $x-y\in \mathbb{Q} $.
Now the set $Y$ is the set of equivalence classes generated by $\sim$. How do we know that a particular equivalence class $[x]$ is countable?
Hint. If $a$ is a representative of the class $[x]$, then observe that $$ [x]=\{a+q : q\in\mathbb Q\}. $$ Hence $|[x]|=\aleph_0$.