# Thread: Intersection of Uncountable Sets Proof

1. ## Intersection of Uncountable Sets Proof

I have a set theory question that I'm not really sure how to do it.

Prove or disprove that there exists an uncountable set $S\subseteq\mathbb{R}$ such that $S\cap(\mathbb{R}\setminus\mathbb{Q})$ is countable.

Thanks!

2. ## Re: Intersection of Uncountable Sets Proof

'u' for binary union, and 'n' for binary intersection:

Suppose S is an uncountable subset of R.

S = S\Q u SnQ.

SnQ is a subset of Q, and Q is countable, so SnQ is countable.

If S\Q were countable, then S would be the union of two countable sets, so S would be countable.

So S\Q is uncountable.

Since S is a subset of R, we have S\Q = S n (R\Q).

So S n (R\Q) is uncountable.

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The key theorem used is that the union of two countable sets is a countable set.