Counting equivalence classes. [duplicate]

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Let $$X$$ denote the set of real transcendental numbers. Define the relation $$\sim$$ on $$X$$ by $$x\sim y$$ iff $$x-y \in \mathbb{Q}$$.

Let $$Y$$ denote the set of equivalence classes generated by $$\sim$$ defined above. Prove that the set $$Y$$ is uncountable.

I tried to do this by contradiction but really haven't gotten anywhere:

Suppose $$Y$$ is countable. Then there exists a bijection $$f:\mathbb{N} \rightarrow Y$$. An equivalence class generated by $$\sim$$ above has the form $$[x] = \{y \in X|y-x \in \mathbb{Q}\}$$. I know I have to somehow show that $$f$$ isn't a bijection, so I could show it isn't surjective/injective, but I am not really sure how to do that. Any hints on how to continue?