Rule of Association/Disjunction/Formulation 1/Proof by Truth Table
< Rule of Association | Disjunction | Formulation 1(Redirected from Rule of Association/Disjunction/Formulation 1/Proof 2)
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Theorem
- $p \lor \paren {q \lor r} \dashv \vdash \paren {p \lor q} \lor r$
Proof
We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.
$\begin{array}{|ccccc||ccccc|} \hline p & \lor & (q & \lor & r) & (p & \lor & q) & \lor & r \\ \hline \F & \F & \F & \F & \F & \F & \F & \F & \F & \F \\ \F & \T & \F & \T & \T & \F & \F & \F & \T & \T \\ \F & \T & \T & \T & \F & \F & \T & \T & \T & \F \\ \F & \T & \T & \T & \T & \F & \T & \T & \T & \T \\ \T & \T & \F & \F & \F & \T & \T & \F & \T & \F \\ \T & \T & \F & \T & \T & \T & \T & \F & \T & \T \\ \T & \T & \T & \T & \F & \T & \T & \T & \T & \F \\ \T & \T & \T & \T & \T & \T & \T & \T & \T & \T \\ \hline \end{array}$
$\blacksquare$