I’m doing some exercises and I want to show an example of a set that can be subtracted from uncountably infinite set and the result will be countably infinite set. I’m a little bit suspicious that there exist such a set. If it is true, can someone give an example of a set?
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1I assume you mean to get a countably infinite set? Otherwise you could trivially subtract the same set from itself. In this case, as Hans says, $\Bbb R-\Bbb R\backslash\Bbb Q$ works. – John Doe Oct 29 '17 at 12:17
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@JohnDoe I almost put that as an answer till I noticed you'd already posted it as a comment. Or (for a finite non-empty set) choose one element of the original and subtract the rest. – Mark Bennet Oct 29 '17 at 12:26
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Yes, I actually mean countably infinite. – user8083314 Oct 29 '17 at 12:40
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As others have said, if you place no restrictions on the set to be removed then it is obviously possible. For any set, subtract the whole set from itself to get the empty set. For a more concrete and slightly less trivial example, subtract the irrational numbers from the reals the get the rationals.
If you only allow a countable set to be subtracted then it is clearly not possible. Even if you allow countably many countable sets to be removed it is still not possible. See Countable union of countable sets is countable.
Of course, if you allow an uncountable collection of countable sets to be subtracted then it becomes possible again in a trivial way.
