Product of Cardinals is Commutative

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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

$\blacksquare$


Sources