Set Difference of Complements

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Theorem

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


Proof

\(\displaystyle \complement \left({S}\right) \setminus \complement \left({T}\right)\) \(=\) \(\displaystyle \left\{ {x: x \in \complement \left({S}\right) \land x \notin \complement \left({T}\right)}\right\}\) Definition of Set Difference
\(\displaystyle \) \(=\) \(\displaystyle \left\{ {x: x \notin S \land x \in T}\right\}\) Definition of Complement
\(\displaystyle \) \(=\) \(\displaystyle \left\{ {x: x \in T \land x \notin S}\right\}\) Rule of Commutation
\(\displaystyle \) \(=\) \(\displaystyle T \setminus S\) Definition of Set Difference

$\blacksquare$