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

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Definition

The disjunction operator is left distributive over the conjunction operator:

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


This can be expressed as two separate theorems:

Forward Implication

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

Reverse Implication

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


Proof by Truth Table

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.

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

$\blacksquare$


Sources

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