Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 2/Proof by Truth Table

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Theorem

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


Proof

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, the truth values under the main connective is true for all boolean interpretations.

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

$\blacksquare$


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