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

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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

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


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.

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

$\blacksquare$


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