Cardinality Less One

Theorem

Let $\left|{S}\right| = n + 1$, where $\left|{S}\right|$ is the cardinality of the finite set $S$.

Let $a \in S$.

Then:

$\left|{S \setminus \left\{{a}\right\}}\right| = n$

where $\setminus$ denotes set difference.

Proof

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

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 17$: Theorem $17.4$