1. Cardinality properties

Hello,

$A_{1}$ and $B_{1}$ have the same cardinality, so do $A_{2}$ and $B_{2}$

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

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

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

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

Counterexample:

$A_{1}={1,2,3}$
$A_{2}={1,2}$
$B_{1}={4,5,6}$
$B_{2}={4,10}$

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

Counterexample:

$A_{1}={1,2,3}$
$A_{2}={1,2}$
$B_{1}={4,5,6}$
$B_{2}={4,10}$

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

2. Re: Cardinality properties

Originally Posted by MachinePL1993
$A_{1}$ and $B_{1}$ have the same cardinality, so do $A_{2}$ and $B_{2}$
I need to check if the following sets also have the same cardinality:
1) $A_{1} \times A_{2}$ and $B_{1} \times B_{2}$.
Those two cartesian products have the same cardinality, as it's easy to construct a bijection between those two sets using bijections $f:A_{1} \rightarrow B_{1}$ and $f:A_{2} \rightarrow B_{2}$.

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

Counterexample:

$A_{1}={1,2,3}$
$A_{2}={1,2}$
$B_{1}={4,5,6}$
$B_{2}={4,10}$

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

Counterexample:

$A_{1}={1,2,3}$
$A_{2}={1,2}$
$B_{1}={4,5,6}$
$B_{2}={4,10}$

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

You have several notation mistakes. It should be
construct a bijection between those two sets using bijections $f:A_{1} \rightarrow B_{1}$ and $g:A_{2} \rightarrow B_{2}$.

And
$A_{1}={1,2,3}$
$A_{2}={1,2}$
$B_{1}={4,5,6}$
$B_{2}={4,10}$

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