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If $A$ and $B$ are infinite sets where $card(A) = card(B)$, what is $card(A\setminus B)$, where $A\setminus B$ denotes relative complement?

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Hope you don't mind, I added some TeX and improved the notation a little. Feel free to rollback if it's not ok. –  Daniel Rust Sep 12 '13 at 16:49
    
It could be a whole lot of things. –  André Nicolas Sep 12 '13 at 16:49

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There is not enough information. If $A=B=\mathbb{N}$ then $A\setminus B=\emptyset$ and so $card(A\setminus B)=0$. If $A=\mathbb{Z}$ and $B=\{n\in A\mid n\geq 0\}$ then $A\setminus B=\{n\in A\mid n\leq -1\}$ and so $card(A\setminus B)=\aleph_0$. What we can say is that $card(A\setminus B)$ is no greater than $card(A)$, but not much more without more information.

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(Assuming AC) We can say that the possible cardinalities are precisely those no greater than $\mathrm{card}(A)$. –  Jonas Meyer Sep 12 '13 at 17:24
    
Thank you this is very good. If A = {0, 1,2,..} = whole numbers and B ={2, 3, 4, ...}, hence, CARD(B) = Alpha null, how would one then calculate the A\B and CARD(A\B)? –  tf177 Sep 12 '13 at 18:20
    
@tf177: For the concrete example you just gave, reviewing the definition of $A\setminus B$ should allow you determine what numbers are in that set. Then you will see how many numbers are in the set. –  Jonas Meyer Sep 13 '13 at 2:51
    
So if set A = {0, 1, 2, 3,...} and B = (2, 4, 6,...}, then B/A = {0, 1, 3,...} = {non-negative odd number}. CARD(B\A) = Alepha null. –  tf177 Sep 13 '13 at 15:12
    
What texts in set theory can I study to learn more about the type of question I posed earlier? –  tf177 Sep 13 '13 at 15:59

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