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

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

Reverse Implication

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

Proof by Truth Table

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:

 $\ds p \land \paren {p \lor q}$ $=$ $\ds \paren {p \lor \bot} \land \paren {p \lor q}$ Disjunction with Contradiction $\ds$ $=$ $\ds p \lor \paren {\bot \land q}$ Disjunction is Left Distributive over Conjunction $\ds$ $=$ $\ds p \lor \bot$ Conjunction with Contradiction $\ds$ $=$ $\ds p$ Disjunction with Contradiction

$\blacksquare$