Cardinality properties

**MachinePL1993** · December 30th 2012, 08:47 AM

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

**Plato** · December 30th 2012, 09:23 AM

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 $|B_{1} \cap B_{2} |=1$ .

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