equivalences

Commutative:

\begin{array}{r} P \land Q \equiv Q \land P \\ P \lor Q \equiv Q \lor P \end{array}

Distributive:

\begin{array}{r} P \land (Q \lor R) \equiv (P \land Q) \lor (P \land R) \\ P \lor (Q \land R) \equiv (P \lor Q) \land (P \lor R) \end{array}

Associative:

\begin{array}{r} P \land (Q \land R) \equiv (P \land Q) \land R \\ P \lor (Q \lor R) \equiv (P \lor Q) \lor R \end{array}

Contrapositive:

\begin{matrix} (3) & \begin{matrix} P \Rightarrow Q \equiv (\neg Q) \Rightarrow (\neg P) \end{matrix} \end{matrix}

DeMorgan's Laws:

\begin{array}{r} \neg (P \land Q) \equiv (\neg P) \lor (\neg Q) \\ \neg (P \lor Q) \equiv (\neg P) \land (\neg Q) \end{array}

Negating Quantifiers:

\begin{array}{r} \neg (\forall x \in X, P (x)) \equiv \exists x \in X, \neg P (x) \\ \neg (\exists x \in X, P (x)) \equiv \forall x \in X, \neg P (x) \end{array}

Negating conditionals:

\begin{matrix} (3) & \neg (P \Rightarrow Q) \equiv P \land \neg Q \end{matrix}