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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2This is not true in general. We need more conditions on $B$ and $C$. At the extreme, $B$ and $C$ could be empty! Even if we ask that the union of $B$ and $C$ be $A$, one of $B$ or $C$ could be tiny. – 2012-04-09
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3But if $A = \mathbb N, B = \{1\}, C = A \setminus B$ you have a counterexample. – 2012-04-09
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0Sorry I should edit the question, because I need to prove that there exist such sets like B and C – 2012-04-09
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0Like $A= \mathbb N$, $B=$ the even numbers and $C=$ the odd numbers? – 2012-04-09
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0Yes but more in general... – 2012-04-09
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0@LePressentiment: As with many of your recent [necrobumps](http://www.urbandictionary.com/define.php?term=necrobump), the title misrepresents the actual question. – 2013-11-06
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0@AsafKaragila: My goal was to add detail to the title, improvements of which by anyone I welcome, and not that for a necrobump. I'd have opted to to edit without causing a bump. "Necrobump" sounds exceptionable in this context, your use of which suggests potential discord between us. Please let me know; I'd love to resolve it! – 2013-11-07
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0@LePressentiment: For the past, I don't know how long, you've been posting edits and answers to posts which are over six months old. One cannot help but conclude that you are gaming for an Archaeologist badge. Some of your edits are reasonable, but others either introduce mistakes or have little to no effect on the actual post; some of the answers that you have posted are either reiteration of other answers, or include mistakes. It's fine to do these things when a post appears on the front page, or is a few days old. It's **NOT FINE** when almost all the posts you "targeted" are year+ old. – 2013-11-07
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0@AsafKaragila: Many thanks for your reply. I tried to address all this in http://meta.math.stackexchange.com/questions/10598/more-precise-question-titles-and-tags?lq=1. Above all, the goal is *not* to "game for an Archaeologist badge." Please understand that there is no target of dates. I try to help improve all posts to my awareness, some of which are older. Please do let me know of other concerns! – 2013-11-07
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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
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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