# Absorption Laws (Logic)/Conjunction Absorbs Disjunction

## Theorem

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

This can be expressed as two separate theorems:

### Forward Implication

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

### Reverse Implication

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

## Proof 1

We apply the Method of Truth Tables.

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

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

$\blacksquare$

## Proof 2

By calculation:

 $\displaystyle p \land \left({p \lor q}\right)$ $=$ $\displaystyle \left({p \lor \bot}\right) \land \left({p \lor q}\right)$ Disjunction with Contradiction $\displaystyle$ $=$ $\displaystyle p \lor \left({\bot \land q}\right)$ Disjunction is Left Distributive over Conjunction $\displaystyle$ $=$ $\displaystyle p \lor \bot$ Conjunction with Contradiction $\displaystyle$ $=$ $\displaystyle p$ Disjunction with Contradiction

$\blacksquare$