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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?

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marked as duplicate by Martín Vacas Vignolo, drhab, Community Dec 29 '18 at 14:53

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