I am attempting to prove or come up with a counterexample for the following statement:
If $|A\cup A'|=\aleph_\alpha$, then either $|A|=\aleph_\alpha$ or $|A'|=\aleph_\alpha$.
Here, $\aleph_\alpha$ are the so-called aleph numbers.
This seems true at least for $\alpha=0$ (the union of two finite sets cannot be countably infinite), but obviously there could be some other $\alpha$ for which this is false. Any hint would be helpful.
Let $|A|=\aleph_\beta$ and $|A'\setminus A|=\aleph_\gamma$. Then $|A\cup A'|=\aleph_\beta+\aleph_\gamma$, so $\max\{\beta,\gamma\}=\alpha$. If $\beta=\alpha$, you’re done, and if $\gamma=\alpha$, there’s just a little more work to be done.