Set Complement inverts Subsets/Proof 4

Theorem

$S \subseteq T \iff \map \complement T \subseteq \map \complement S$


Proof

\(\ds S\) \(\subseteq\) \(\ds T\)
\(\ds \leadstoandfrom \ \ \) \(\ds S\) \(=\) \(\ds S \cup T\) Union with Superset is Superset‎
\(\ds \leadstoandfrom \ \ \) \(\ds \map \complement S\) \(=\) \(\ds \map \complement {S \cup T}\) Complement of Complement
\(\ds \) \(=\) \(\ds \map \complement S \cap \map \complement T\) De Morgan's Laws: Complement of Union
\(\ds \leadstoandfrom \ \ \) \(\ds \map \complement S\) \(\subseteq\) \(\ds \map \complement T\) Intersection with Subset is Subset

$\blacksquare$

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