I'm having a hard time with this proof.
The definition of cardinality that I'm using is: Two sets have the same cardinality if there is a bijection between them.
($\star$) I already prove that $|B\setminus A|$ is an infinite set.
I'm doing a proof by contradiction. So I'm assuming that $|B|\neq|B\setminus A|$ wich means that there is no bijection between $B$ and $A\setminus B$, what I'm trying to do is that this assumption leads me to say that $|A\setminus B|$ is finite which will contradict $(\star)$.
But I don't know how the assumption will let me say that $|A\setminus B|$ is finite.
I already try using the fact that each infinite set has a countable subset, but I didn't succeed.
Any ideas for this?
Thank you.
Best Answer
Every infinite set has an infinite countable subset (assuming AC). So $B\setminus A=C\uplus D$ where $C$ is countable and infinite and the union is disjoint. Then $B\setminus A=|C|+|D|=\aleph_0+|D|$. Also $B=A\uplus C\uplus D$, so $|B|=|A|+\aleph_0+|D|$. As $A$ is finite, $|A|+\aleph_0=\aleph_0$.