Identity of Cardinal Sum is Zero

Theorem

Let $\mathbf a$ be a cardinal.

Then:

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

where $\mathbf a + \mathbf 0$ denotes the sum of the zero cardinal and $\mathbf a$.

That is, $\mathbf 0$ is the identity element of the sum operation on cardinals.

Proof

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

From Union with Empty Set we have $A \cup \O = A$.

From Intersection with Empty Set we have $A \cap \O = \O$.

So $A$ and $\O$ are disjoint and so:

\(\ds \mathbf a + \mathbf 0\)

\(=\)

\(\ds \card {A \cup \O}\)

Definition of Sum of Cardinals

\(\ds \)

\(=\)

\(\ds \card A + \card \O\)

Cardinality of Set Union/Corollary

\(\ds \)

\(=\)

\(\ds \card A\)

Cardinality of Empty Set

\(\ds \)

\(=\)

\(\ds \mathbf a\)

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 8$: Theorem $8.5: \ (3)$