Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 2

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Theorem

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


Forward Implication

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


Reverse Implication

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


Proof 1

By the tableau method of natural deduction:

$\vdash \left({p \lor \left({q \land r}\right)}\right) \iff \left({\left({p \lor q}\right) \land \left({p \lor r}\right)}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $p \lor \left({q \land r}\right)$ Assumption (None)
2 1 $\left({p \lor q}\right) \land \left({p \lor r}\right)$ Sequent Introduction 1 Disjunction is Left Distributive over Conjunction: Formulation 1
3 $\left({p \lor \left({q \land r}\right)}\right) \implies \left({\left({p \lor q}\right) \land \left({p \lor r}\right)}\right)$ Rule of Implication: $\implies \mathcal I$ 1 – 2 Assumption 1 has been discharged
4 4 $\left({p \lor q}\right) \land \left({p \lor r}\right)$ Assumption (None)
5 4 $p \lor \left({q \land r}\right)$ Sequent Introduction 4 Disjunction is Left Distributive over Conjunction: Formulation 1
6 $\left({\left({p \lor q}\right) \land \left({p \lor r}\right)}\right) \implies \left({p \lor \left({q \land r}\right)}\right)$ Rule of Implication: $\implies \mathcal I$ 4 – 5 Assumption 4 has been discharged
7 $\left({p \lor \left({q \land r}\right)}\right) \iff \left({\left({p \lor q}\right) \land \left({p \lor r}\right)}\right)$ Biconditional Introduction: $\iff \mathcal I$ 3, 6

$\blacksquare$


Proof 2

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, the truth values under the main connective is true for all boolean interpretations.

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

$\blacksquare$


Sources

(erroneously referring to it as one of De Morgan's Laws)