Countable subsets of an uncountable set

Question

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?

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

score 4 · Accepted Answer · answered Oct 20 '12 at 18:21

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.

answered Oct 20 '12 at 18:21

Brian M. Scott

631,399

Michael Hardy · Answer 2 · 2020-06-21T19:44:18.550

1

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

edited Jun 21 '20 at 19:44

answered Oct 20 '12 at 22:01

Michael Hardy

1

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

user3555551 · Answer 3 · 2021-07-11T07:56:53.127

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.

8 years later... I suppose my answer will help someone else down the road, as I aim to prove your belief that if $X$ is uncountable and $A \subset X$ is countable, then $A^C$ is uncountable.

I am assuming you've already proven that the union of countable sets is countable. Others have already given you excellent counterexamples to your question.

Suppose that $A^C$ is countable, as is $A$ by definition. Then $A^C \cup A = X$ is countable. But $X$ is uncountable which we defined. So $A^C$ must be uncountable.

Hope this solidifies your belief.

Countable subsets of an uncountable set

3 Answers3