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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?

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Also relevant: math.stackexchange.com/questions/17432 –  Asaf Karagila Oct 20 '12 at 18:40

2 Answers 2

up vote 4 down vote accepted

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.

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

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I never knew that "numbers" is synonymous with "real numbers". –  Asaf Karagila Oct 20 '12 at 22:03
    
@AsafKaragila : Neither did I. Why would you say such a thing? –  Michael Hardy Oct 20 '12 at 22:04
    
Because the set of non-negative numbers could also mean $\{k\in\mathbb Z\mid k\geq 0\}$; or in $\mathbb Q$; or in the real closure of $\mathbb Q$; or in the computable numbers; or so on and so forth. –  Asaf Karagila Oct 20 '12 at 22:06
    
But I think the context makes clear what I'm talking about. –  Michael Hardy Oct 20 '12 at 22:22
    
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"). –  Asaf Karagila Oct 20 '12 at 22:40

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