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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?

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    Hint: $[x] = \{ z \mid \exists q \in \Bbb{Q}\cdot z = x + q\}$. – 2017-01-07
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    Typing note: `$~$` is used to increase space between objects while `\!` is used to decrease space between objects. E.g. `$1~~~2\!3$` produces $1~~~2\!3$. To have the symbol `~` appear as a symbol, use instead `$\sim$` to produce $\sim$. – 2017-01-07

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Hint. If $a$ is a representative of the class $[x]$, then observe that $$ [x]=\{a+q : q\in\mathbb Q\}. $$ Hence $|[x]|=\aleph_0$.