Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 1/Proof by Truth Table

Theorem

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

Proof
We apply the Method of Truth Tables to the proposition.

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

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