Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 2/Proof by Truth Table
Theorem
- $\vdash \paren {p \lor \paren {q \land r} } \iff \paren {\paren {p \lor q} \land \paren {p \lor r} }$
Proof
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
- 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}$: Exercise $14 \ \text{(c)}$
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.4$: Statement Forms: Exercise $\text{II}. \ 7$