Absorption Laws (Logic)

Theorem

 * $$P \land \left({P \lor Q}\right) \vdash P$$
 * $$P \vdash P \land \left({P \lor Q}\right)$$


 * $$P \lor \left ({P \land Q}\right) \vdash P$$
 * $$P \vdash P \lor \left ({P \land Q}\right)$$

These are called the Absorption Laws or Absorption Identities.

Proof

 * $$P \land \left({P \lor Q}\right) \vdash P$$:


 * $$P \vdash P \land \left({P \lor Q}\right)$$:


 * $$P \lor \left ({P \land Q}\right) \vdash P$$


 * $$P \vdash P \lor \left ({P \land Q}\right)$$