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 $\displaystyle S\subseteq\mathbb{R}$ such that $\displaystyle S\cap(\mathbb{R}\setminus\mathbb{Q})$ is countable.
Thanks!
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 $\displaystyle S\subseteq\mathbb{R}$ such that $\displaystyle S\cap(\mathbb{R}\setminus\mathbb{Q})$ is countable.
Thanks!
'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.
/
The key theorem used is that the union of two countable sets is a countable set.