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I was wondering about exactly what is in the title:

Is it true that for every subset $A\subseteq\Bbb R$: $$\Bbb R\sim A\text{ or }\Bbb R\sim \Bbb R\setminus A$$ where $\sim$ denotes equinumerosity

I would myself say this is the case, however I have not been able to prove this. I could however think of some cases that involve the continuum hypothesis in order to work, or isn't this the case?

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We use Lemma 1: If $U\cup V$ is infinite, then $|U\cup V|=\max(|U|,|V|)$.

Apply this to $A$ and $\mathbb R\setminus A$.

This does not require the continuum hypothesis, but it does require axiom of choice. Specifically, Lemma 1 follow from the fact that if $U$ is infinite then $|2\cdot U|=|U|$, which is often proven using Zorn's lemma, which is equivalent to the Axiom of Choice.

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