Rule of Distribution/Disjunction Distributes over Conjunction/Right Distributive/Formulation 1
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.
$\begin{array}{|ccccc||ccccccc|} \hline (q & \land & r) & \lor & p & (q & \lor & p) & \land & (r & \lor & p) \\ \hline F & F & F & F & F & F & F & F & F & F & F & F \\ F & F & F & T & T & F & T & T & T & F & T & T \\ F & F & T & F & F & F & F & F & F & T & T & F \\ F & F & T & T & T & F & T & T & T & T & T & T \\ T & F & F & F & F & T & T & F & F & F & F & F \\ T & F & F & T & T & T & T & T & T & F & T & T \\ T & T & T & T & F & T & T & F & T & T & T & F \\ T & T & T & T & T & T & T & T & T & T & T & T \\ \hline \end{array}$
$\blacksquare$