Cardinality Less One

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Theorem

Let $S$ be a finite set.

Let:

$\left\lvert{S}\right\rvert = n + 1$

where $\left\lvert{S}\right\rvert$ is the cardinality of $S$.

Let $a \in S$.


Then:

$\left\lvert{S \setminus \left\{ {a}\right\} }\right\rvert = n$

where $\setminus$ denotes set difference.


Proof

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

$\blacksquare$


Sources