De Morgan's Laws

Proposition 1. De Morgan’s Laws.

Let $X$ be a set. For any $A \subseteq X$ write

\[\overline{A} = X \setdif A.\]

For any $A$, $B \subseteq X$, we have:

1. \[\overline{A \cup B} = \overline{A} \cap \overline{B}.\]
2. \[\overline{A \cap B} = \overline{A} \cup \overline{B}.\]

More generally, for any $A\subseteq\powerset(X)$:

3. \[\overline{\bigcup A} = \bigcap_{Y \in A} \overline{Y}.\]
4. \[\overline{\bigcap A} = \bigcup_{Y \in A} \overline{Y}.\]