# Cardinality of complement of uncountable subset of uncountable set. [closed]

We have a set S which is uncountably infinite. U is a subset of S and it is uncountably infinite.

What is the cardinality of the complement of U with respect to S?

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## closed as too localized by François G. Dorais♦, Harald Hanche-Olsen, Pete L. Clark, S. Carnahan♦, Kevin H. LinApr 18 '10 at 20:35

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MO is the wrong place for this. Perhaps a homework problem? So, read the FAQ here (see the link at the top?) before posting... –  Gerald Edgar Apr 18 '10 at 13:04
I removed the complexity-theory tag, since this question does not involve complexity theory. –  Joel David Hamkins Apr 18 '10 at 20:03

Take $S=\mathbb{R}$. If you take $U$ to be $\mathbb{R} \setminus \{ 1 \}$. Then the complement of $U$ in $S$ has cardinality 1. If you take $U=\mathbb{R} \setminus \mathbb{Q}$ then the complement is countably infinite. If you take $U$ to be the set of numbers which are positive, then the complement is uncountably infinite.