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

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Theorem

The disjunction operator is right distributive over the conjunction operator:

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


This can be expressed as two separate theorems:

Forward Implication

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

Reverse Implication

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


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.

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

$\blacksquare$