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

Theorem

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

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.

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

$\blacksquare$