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

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Definition

The conjunction operator is left distributive over the disjunction operator:

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


This can be expressed as two separate theorems:

Forward Implication

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

Reverse Implication

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


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.

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

$\blacksquare$


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