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 of Finite Sets/Corollary

\(\ds \)

\(=\)

\(\ds \mathbf b \mathbf a\)

Definition of Product of Cardinals

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 8$: Theorem $8.4: \ (1)$