ThmDex – An index of mathematical definitions, results, and conjectures.

Result R1036 on D80: Power set

Power set is closed under complements

Formulation 0

Let $X$ be a D11: Set such that

(i)	$\mathcal{P}(X)$ is the D80: Power set of $X$
(ii)	$E \in \mathcal{P}(X)$

Then \begin{equation} X \setminus E \in \mathcal{P}(X) \end{equation}

Formulation 1

Let $X$ be a D11: Set.

Then \begin{equation} \forall \, E \in \mathcal{P}(X) : E^{\complement} \in \mathcal{P}(X) \end{equation}

Formulation 2

Let $X$ be a D11: Set.

Then \begin{equation} \forall \, E \subseteq X : X \setminus E \subseteq X \end{equation}

Formulation 3

Let $X$ be a D11: Set.

Then \begin{equation} \forall \, E \subseteq X : E^{\complement} \subseteq X \end{equation}

Proofs

Proof 0

Let $X$ be a D11: Set such that

(i)	$\mathcal{P}(X)$ is the D80: Power set of $X$
(ii)	$E \in \mathcal{P}(X)$