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?
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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You cannot say anything about the cardinality of the complement. It can either be finite, countable infinite and uncountable infinite. The following examples may illustrate this. 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. 

