# Product of Cardinals is Commutative

## Theorem

Let $\mathbf a$ and $\mathbf b$ be cardinals.

Then:

$\mathbf a \mathbf b = \mathbf b \mathbf a$

where $\mathbf a \mathbf b$ denotes the product of $\mathbf a$ and $\mathbf b$.

## Proof

Let $\mathbf a = \card A$ and $\mathbf b = \card B$ for some sets $A$ and $B$.

Then:

 $\ds \mathbf a \mathbf b$ $=$ $\ds \card {A \times B}$ Definition of Product of Cardinals $\ds$ $=$ $\ds \card {B \times A}$ Cardinality of Cartesian Product/Corollary $\ds$ $=$ $\ds \mathbf b \mathbf a$ Definition of Product of Cardinals

$\blacksquare$