Complements Invert Subsets

From ProofWiki

Jump to: navigation, search

Theorem

$S \subseteq T \iff \complement \left({T}\right) \subseteq \complement \left({S}\right)$

where:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle S$	$\subseteq$	$\displaystyle T$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle S \cap T$	$=$	$\displaystyle S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Intersection with Subset is Subset‎
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle \complement \left({S \cap T}\right)$	$=$	$\displaystyle \complement \left({S}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Relative Complement of Relative Complement
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle \complement \left({S}\right) \cup \complement \left({T}\right)$	$=$	$\displaystyle \complement \left({S}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	De Morgan's Laws
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle \complement \left({T}\right)$	$\subseteq$	$\displaystyle \complement \left({S}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Union with Superset is Superset

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle S\)	\(\subseteq\)	\(\displaystyle T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle S \cap T\)	\(=\)	\(\displaystyle S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Intersection with Subset is Subset‎
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \complement \left({S \cap T}\right)\)	\(=\)	\(\displaystyle \complement \left({S}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Relative Complement of Relative Complement
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \complement \left({S}\right) \cup \complement \left({T}\right)\)	\(=\)	\(\displaystyle \complement \left({S}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	De Morgan's Laws
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \complement \left({T}\right)\)	\(\subseteq\)	\(\displaystyle \complement \left({S}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Union with Superset is Superset