# Absorption Laws (Logic)/Disjunction Absorbs Conjunction

## Theorem

$p \lor \paren {p \land q} \dashv \vdash p$

This can be expressed as two separate theorems:

### Forward Implication

$p \lor \paren {p \land q} \vdash p$

### Reverse Implication

$p \vdash p \lor \paren {p \land q}$

## Proof 1

We apply the Method of Truth Tables.

As can be seen by inspection, the appropriate truth values match for all boolean interpretations.

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

$\blacksquare$

## Proof 2

 $\ds p \lor \left({p \land q}\right)$ $=$ $\ds \left({p \land \top}\right) \lor \left({p \land q}\right)$ Conjunction with Tautology $\ds$ $=$ $\ds p \land \left({\top \lor q}\right)$ Conjunction is Left Distributive over Disjunction $\ds$ $=$ $\ds p \land \top$ Disjunction with Tautology $\ds$ $=$ $\ds p$ Conjunction with Tautology

$\blacksquare$