# Algebra of cardinality

## Other

### Cardinality

#### Cardinality of cartesian product

What about the cardinality of Cartesian products? So if we have sets:

\(\{1,2,3\}\)

\(\{a,b\}\)

We can have the Cartesian product set:

\(\{(1,a),(2,a),(3,a),(1,b),(2,b),(3,b)\}\)

We can see that:

\(|A.B|=|A|.|B|\)

#### Cardinality of union and intersection

\(|A\lor B| = |A|+|B|-|A\land B|\)

#### Cardinality of powerset

\(|P(s)|=2^{|s|}\)

#### Cardinality of complement

\(|a \setminus b|=|a|-|a\land b|\)

#### Cardinality of even/odd natural numbers

What about the cardinality of even numbers? Well, we can define a bijective function between each:

\(f(n)=2n\)

Similarly for odd numbers:

\(f(n)=2n+1\)

So these both have cardinality \(\aleph_0\).