Set Complement inverts Subsets/Corollary

From ProofWiki
Jump to navigation Jump to search

Corollary to Set Complement inverts Subsets

Let $S$ and $T$ be sets.

Then:

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

where:

$S \subseteq \map \complement T$ denotes that $S$ is a subset of the set complement of $T$.


Proof

\(\ds S\) \(\subseteq\) \(\ds \map \complement T\)
\(\ds \leadstoandfrom \ \ \) \(\ds \map \complement {\map \complement T}\) \(\subseteq\) \(\ds \map \complement S\) Set Complement inverts Subsets
\(\ds \leadstoandfrom \ \ \) \(\ds T\) \(\subseteq\) \(\ds \map \complement S\) Complement of Complement

$\blacksquare$


Sources