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

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Theorem

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


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}$

$\blacksquare$

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