# What is the cardinality of the relative complement of two infinite sets that have the same cardinality?

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. – Dan Rust Sep 12 '13 at 16:49
It could be a whole lot of things. – André Nicolas Sep 12 '13 at 16:49

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.
(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
@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