Absorption Laws (Logic)

Theorem
For any two propositions $p$ and $q$, we have:


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

These are called the Absorption Laws or Absorption Identities.

Their abbreviation in a tableau proof is $\mathrm {AL}$.

Proof by Tableau
Proceed 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.

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

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