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 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 $8.4: \ (1)$