Power Set of Empty Set

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Theorem

The power set of the empty set $\varnothing$ is the set $\left\{{\varnothing}\right\}$.


Proof

From Empty Set is Element of Power Set and Set is Element of its Power Set:

$\varnothing \in \mathcal P \left({\varnothing}\right)$

From Empty Set is Subset of All Sets:

$S \subseteq \varnothing \implies S = \varnothing$

That is:

$S \in \mathcal P \left({\varnothing}\right) \implies S = \varnothing$

Hence the only element of $\mathcal P \left({\varnothing}\right)$ is $\varnothing$.

$\blacksquare$


Sources