Let $X$ be a set and write
\[\overline{A} = X\setminus A.\]For any $A$, $B\subseteq X$:
- \[\overline{A\cup B} = \overline{A}\cap\overline{B}.\]
- \[\overline{A\cap B} = \overline{A}\cup\overline{B}.\]
More generally, for any $A\subseteq\powerset(X)$:
- \[\overline{\bigcup A} = \bigcap_{Y\in A}\overline{Y}.\]
- \[\overline{\bigcap A} = \bigcup_{Y\in A}\overline{Y}.\]