Logic
DeMorgan's laws
$ \lnot(P \land Q) \Leftrightarrow \lnot P \lor \lnot Q
$ \lnot(P \lor Q) \Leftrightarrow \lnot P \land \lnot Q
Commutative laws
$ P \land Q \Leftrightarrow Q \land P
$ P \lor Q \Leftrightarrow Q \lor P
Associative laws
$ P \land (Q \land R) \Leftrightarrow (P \land Q) \land R
$ P \lor (Q \lor R) \Leftrightarrow (P \lor Q) \lor R
Idempotent laws
$ P \land P \Leftrightarrow P
$ P \lor P \Leftrightarrow P
Distributive laws
$ P \land (Q \lor R) \Leftrightarrow (P \land Q ) \lor (P \land R)
$ P \lor (Q \land R) \Leftrightarrow (P \lor Q ) \land (P \lor R)
Absorption laws
$ P \land (P \lor Q) \Leftrightarrow P
$ P \lor (P \land Q) \Leftrightarrow P
Double negation law
$ \lnot\lnot P \Leftrightarrow P
Tautology laws
$ P \land (a\ tautology) \Leftrightarrow P
$ P \lor (a\ tautology) \Leftrightarrow a\ tautology
$ \lnot (a\ tautology) \Leftrightarrow a\ contradiction
Contradiction laws
$ P \land (a\ contradiction) \Leftrightarrow a\ contradiction
$ P \lor (a\ contradiction) \Leftrightarrow P
$ \lnot (a\ contradiction) \Leftrightarrow a\ tautology
Conditional law
$ P \rightarrow Q \Leftrightarrow \lnot P \lor Q
$ P \rightarrow Q \Leftrightarrow \lnot (P \land \lnot Q)
Contrapositive law
$ P \rightarrow Q \Leftrightarrow \lnot Q \rightarrow \lnot P
Quantifier negation laws
$ \lnot \exists P(x) \Leftrightarrow \forall \lnot P(x)
$ \lnot \forall P(x) \Leftrightarrow \exists \lnot P(x)
Quantifier distributive laws
$ \forall x [P(x) \land Q(x)] \Leftrightarrow \forall x P(x) \land \forall x Q(x)
$ \exists x [P(x) \lor Q(x)] \Leftrightarrow \exists x P(x) \lor \exists x Q(x)