De Morgan’s Laws are used in Set Theory, Logic, and several other related fields. Both pairs of equivalencies mean the same, but the two fields use slightly different notation.
Using Notation from Logic we have the following 2 equivalencies:
\thicksim (p \land q) \implies \thicksim p\enspace\lor \thicksim q
\thicksim (p \lor q) \implies \thicksim p\enspace\land \thicksim q
Using Notation from Sets we have the following 2 equivalencies:
(p \cap q)' \implies p' \cup q'
(p \cup q)' \implies p' \cap q'
Both of these pairs can be used to show other equivalencies, or prove statements.
Examples to be added…