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 such that is countable.
'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.