Zero of Cardinal Product is Zero

Theorem

Let $\mathbf a$ be a cardinal.

Then:

$\mathbf 0 \mathbf a = \mathbf 0$

where $\mathbf 0 \mathbf a$ denotes the product of the (cardinal) zero and $\mathbf a$.

That is, $\mathbf 0$ is the zero element of the product operation on cardinals.

Proof

Let $\mathbf a = \card A$ for some set $A$.

From the definition of (cardinal) zero, $\mathbf 0$ is the cardinal associated with the empty set $\O$.

We have by definition of product of cardinals that $\mathbf 0 \mathbf a$ is the cardinal associated with $\O \times A$.

But from Cartesian Product is Empty iff Factor is Empty:

$\O \times A = \O$

Hence the result.

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 8$: Theorem $8.4: \ (4)$