Sum 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 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

$\blacksquare$