Originally Posted by

**MachinePL1993** $\displaystyle A_{1} $ and $\displaystyle B_{1} $ have the same cardinality, so do $\displaystyle A_{2} $ and $\displaystyle B_{2} $

I need to check if the following sets also have the same cardinality:

1)$\displaystyle A_{1} \times A_{2} $ and $\displaystyle B_{1} \times B_{2} $.

Those two cartesian products have the same cardinality, as it's easy to construct a bijection between those two sets using bijections $\displaystyle f:A_{1} \rightarrow B_{1}$ and $\displaystyle f:A_{2} \rightarrow B_{2}$.

2) $\displaystyle A_{1} \cap A_{2} $ and $\displaystyle B_{1} \cap B_{2} $.

Counterexample:

$\displaystyle A_{1}={1,2,3}$

$\displaystyle A_{2}={1,2}$

$\displaystyle B_{1}={4,5,6}$

$\displaystyle B_{2}={4,10}$

Then $\displaystyle |A_{1} \cap A_{2}|=2$ and $\displaystyle |B_{1} \cap B_{2} |=1$.

2) $\displaystyle A_{1} \cap A_{2} $ and $\displaystyle B_{1} \cap B_{2} $.

Counterexample:

$\displaystyle A_{1}={1,2,3}$

$\displaystyle A_{2}={1,2}$

$\displaystyle B_{1}={4,5,6}$

$\displaystyle B_{2}={4,10}$

Then $\displaystyle |A_{1} \cup A_{2}|=3$ and $\displaystyle |B_{1} \cup B_{2} |=4$.