1. ## Cardinality properties

Hello,

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

2. ## Re: Cardinality properties

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 $\displaystyle |B_{1} \cup B_{2} |=4$.

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

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

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