Set Complement inverts Subsets/Proof 1

Theorem

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


Proof

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

$\blacksquare$

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