# Set Difference of Complements

$\complement \left({S}\right) \setminus \complement \left({T}\right) = T \setminus S$
 $\displaystyle \complement \left({S}\right) \setminus \complement \left({T}\right)$ $=$ $\displaystyle \left\{ {x: x \in \complement \left({S}\right) \land x \notin \complement \left({T}\right)}\right\}$ 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
$\blacksquare$