Sum 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 sum of $\mathbf a$ and $\mathbf b$.
Proof
Let $\mathbf a = \map \Card A$ and $\mathbf b = \map \Card B$ for some sets $A$ and $B$ such that $A \cap B = \O$.
Then:
\(\ds \mathbf a + \mathbf b\) | \(=\) | \(\ds \map \Card {A \cup B}\) | Definition of Sum of Cardinals | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Card {B \cup A}\) | Union is Commutative | |||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf b + \mathbf a\) | Definition of Sum of Cardinals |
$\blacksquare$
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 8$: Theorem $8.5: \ (1)$