I'm dealing with a question and I don't know how to formulate my answer. The question is: Let A be an infinite set. Prove that there exist two sets: B and C such that they are subsets of A, and also B and C are pairwise disjoint, then B and C are infinite sets as well.
If $A$ is infinite then it has two infinite subsets $B, C$ which are pairwise disjoint.
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elementary-set-theory
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0@LePressentiment: There are more than enough *new* posts to edit and correct. Moreover, it's fine to edit old posts *on occasion*, but not as a daily routine. It's not fine when the edits themselves introduce inaccuracies. – 2013-11-07
1 Answers
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From your comments it seems you want to prove that there are disjoint subsets $B$ and $C$ of $A$, both of which are infinite. To do this, you could use the
Hint: Let A' be a countably infinite subset of $A$. Let $f$ be a bijection from $\Bbb N$ to A'. Consider the sets $f(\Bbb N_{\rm e})$ and $f(\Bbb N_{\rm o})$, where $\Bbb N_{\rm e}$ is the set of even positive integers and $\Bbb N_{\rm o}$ is the set of odd positive integers.
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0This argument doesn't work if $A$ is Dedekind-finite. I understand the theorem is false in this case (i.e. you need countable choice to prove this)? – 2014-02-05