4
$\begingroup$

I believe it's true that if I have an uncountably infinite set $X$ and a countable subset $A$, then it's complement, $A^c$ is uncountable.

Is it also true that if I have an uncountable subset of $X$, called $B$, the complement of this set, $B^c$, is countable?

  • 0
    Also relevant: http://math.stacke$x$change.com/q$u$estions/17432/2012-10-20

2 Answers 2

4

Not in general, no. For a simple example, consider the uncountable set $[0,2)\subseteq\Bbb R$: it’s the union of the complementary subsets $[0,1)$ and $[1,2)$, which are clearly both uncountable.

0

The set of non-negative numbers is uncountable, and is complement in $\mathbb R$, the set of negative numbers, is also uncountable.

  • 0
    I think that the answer could benefit from a slightly improved choice of words, e.g. "The set of non-negative real numbers" or so. You have a typo anyway to correct ("and **is** complement" to "and **its** complement").2012-10-20