Equivalences

A collection of logical equivalences

1. Absorption: §

$$\begin{array}{rl}P\wedge P& \equiv P\\ P\vee P& \equiv P\end{array}$$

2. Commutativity: §

$$\begin{array}{rl}P\wedge Q& \equiv Q\wedge P\\ P\vee Q& \equiv Q\vee P\end{array}$$

3. Associativity: §

$$\begin{array}{r}P\wedge (Q\wedge R)\equiv (P\wedge Q)\wedge R\\ P\vee (Q\vee R)\equiv (P\vee Q)\vee R\end{array}$$

4. Distributivity: §

$$\begin{array}{rl}P\wedge (Q\vee R)& \equiv (P\wedge Q)\vee (P\wedge R)\\ P\vee (Q\wedge R)& \equiv (P\vee Q)\wedge (P\vee R)\end{array}$$

5. Double negation: §

$$P\equiv \neg \neg P$$

6. De Morgan’s Laws: §

7. Implication: §

$$P\to Q\equiv \neg P\wedge Q$$

8. Contraposition: §

$$\begin{array}{rl}\neg P\to \neg Q& \equiv Q\to P\\ P\to \neg Q& \equiv Q\to \neg P\\ \neg P\to Q& \equiv \neg Q\to P\end{array}$$

9. Bi-implication: §

$$P\leftrightarrow Q\equiv (P\wedge Q)\vee (\neg P\wedge \neg Q)$$

10. Valid and Unsatisfiable: §

$$\begin{array}{rl}P& \equiv \top \text{ when P is valid}\\ P& \equiv \perp \text{ when P is unsatisfiable}\end{array}$$

11. Duality: §

$$\begin{array}{rl}\neg \top & \equiv \perp \\ \neg \perp & \equiv \top \end{array}$$

12. Negation from absurdity: §

$$P\to \perp \equiv \neg P$$

13. Identity: §

$$\begin{array}{rl}P\vee \perp & \equiv P\\ P\wedge \top & \equiv P\end{array}$$

14. Dominance: §

$$\begin{array}{rl}P\vee \top & \equiv \top \\ P\wedge \perp & \equiv \perp \end{array}$$

15. Contradiction: §

$$P\wedge \neg P\equiv \perp$$

16. Excluded Middle: §

$$P\vee \neg P\equiv \top$$