Absorption Laws (Logic)

Theorem

 * $$p \land \left({p \lor q}\right) \dashv \vdash p$$
 * $$p \lor \left ({p \land q}\right) \dashv \vdash p$$

These are called the Absorption Laws or Absorption Identities.

Proof by Natural deduction
These are proved by the Tableau method.

Proof by Truth Table
We apply the Method of Truth Tables to the propositions in turn.

As can be seen by inspection, in all cases the appropriate truth values match for all models.

$$\begin{array}{|ccccc||c|} \hline p & \land & (p & \lor & q) & p \\ \hline F & F & F & F & F & F \\ F & F & F & T & T & F \\ T & T & T & T & F & T \\ T & T & T & T & T & T \\ \hline \end{array}$$

$$\begin{array}{|ccccc||c|} \hline p & \lor & (p & \land & q) & p \\ \hline F & F & F & F & F & F \\ F & F & F & F & T & F \\ T & T & T & F & F & T \\ T & T & T & T & T & T \\ \hline \end{array}$$

Comment
The name absorption laws is also used for the equivalent results in set theory: Union with Intersection and Intersection with Union.