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

Theorem

The conjunction operator is right distributive over the disjunction operator:

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

This can be expressed as two separate theorems:

Forward Implication

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

Reverse Implication

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

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.

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

$\blacksquare$

Sources

• 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Exercise $6 \ \text{(b)}$