# 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 $$\aleph_0$$.