Identity of Cardinal Sum is Zero

From ProofWiki
Jump to navigation Jump to search


Let $\mathbf a$ be a cardinal.


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

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.


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
\(\ds \) \(=\) \(\ds \mathbf a\)