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

Theorem

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

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.

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