De Morgan’s laws describe how mathematical statements and concepts are related through their opposites.
The structure of De Morgan’s laws, whether applied to Sets, Propositions, or logic gates, is always the same.

## Sets

The complement of the union of two sets \(A\) and \(B\) is equal to the intersection of their complements.
\[
\overline{A \cup B} = \overline{A} \cap \overline{B}
\]
The complement of the intersection of two set \(A\) and \(B\) is equal to the union of their complements
\[
\overline{A \cap B} = \overline{A} \cup \overline{B}
\]

## Propositions

\[
\lnot(p \lor q) \equiv \lnot p \land \lnot q
\]
\[
\lnot(p \land q) \equiv \lnot p \lor \lnot q
\]

## Logic

\[
\lnot [(\forall x) P(x)] \equiv (\exists x)[\lnot P(x)]
\]
\[
\lnot [(\exists x) P(x)] \equiv (\forall x)[\lnot P(x)]
\]