Given two sets $A$ and $B$, if I know:
For every element $a\in A$, there is an element $b\in B$ which has the same cardinality as $a$, i.e. $|a| = |b|$, and also for every $b\in B$ there is an element $a\in A$ such that $|a| = |b|$.
Every element in $A$ has infinite cardinality, and every element in $B$ has infinite cardinality.
Can I prove $\bigcup A$ has the same cardinality as $\bigcup B$? If the answer is yes, may I please ask for a proof? If not (which surprises me), how do you slightly modify the condition on infiniteness so we can prove it?
Hints and references to existing posts or theorems are also appreciated. Thank you!
NO. Even if $|A|=|B|$. E.g. if $A$ is the set of all countably infinite subsets of $\Bbb N$ and if $B$ is the set of all countably infinite subsets of $\Bbb R$ then $|A|=|B|=2^{\aleph_0}$. But $|\bigcup A|=|\Bbb N|=\aleph_0<2^{\aleph_0}=|\Bbb R|=|\bigcup B|.$