Rule of Distribution/Disjunction Distributes over Conjunction

Definition
Disjunction distributes over conjunction:

Disjunction is Left Distributive over Conjunction

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

Disjunction is Right Distributive over Conjunction

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

Alternative rendition
These can alternatively be rendered as:

They can be seen to be logically equivalent to the forms above.

Proof
By the tableau method of natural deduction:

Then we use the Rule of Commutation:

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 models.

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

$\left({q \land r}\right) \lor p \dashv \vdash \left({q \lor p}\right) \land \left({r \lor p}\right)$ is demonstrated similarly.