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

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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)	$E \in \mathcal{P}(X)$

$\square$