Symmetric Difference of Complements

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Theorem

The symmetric difference of two sets is the same as the symmetric difference of their complements:

$\complement \left({S}\right) * \complement \left({T}\right) = S * T$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \complement \left({S}\right) * \complement \left({T}\right)$	$=$	$\displaystyle \left({\complement \left({S}\right) \setminus \complement \left({T}\right)}\right) \cup \left({\complement \left({T}\right) \setminus \complement \left({S}\right)}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Symmetric Difference
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left({T \setminus S}\right) \cup \left({S \setminus T}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Set Difference of Complements
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle S * T$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Symmetric Difference

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \complement \left({S}\right) * \complement \left({T}\right)\)	\(=\)	\(\displaystyle \left({\complement \left({S}\right) \setminus \complement \left({T}\right)}\right) \cup \left({\complement \left({T}\right) \setminus \complement \left({S}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Symmetric Difference
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({T \setminus S}\right) \cup \left({S \setminus T}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Set Difference of Complements
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle S * T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Symmetric Difference