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

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

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