Cardinality Less One

Theorem

Let $S$ be a finite set.

Let:

$\card S = n + 1$

where $\card S$ is the cardinality of $S$.

Let $a \in S$.

Then:

$\card {S \setminus \set a} = n$

where $\setminus$ denotes set difference.

Proof

This follows as an immediate consequence of Set Equivalence Less One Element.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 17$: Finite Sets: Theorem $17.4$