Question

Using the truth table, verify \( p \lor (q \land r) \equiv (p \lor q) \land (p \lor r) \).

Hint:

When constructing a truth table for \(n\) variables, there will be \(2^n\) rows. Be systematic in listing all possible combinations of T and F to ensure you don't miss any cases.

Answer 1

We construct a truth table to evaluate both sides of the equivalence. \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline \(p\) & \(q\) & \(r\) & \(q \land r\) & \(p \lor (q \land r)\) & \(p \lor q\) & \(p \lor r\) & \((p \lor q) \land (p \lor r)\)
\hline T & T & T & T & T & T & T & T
T & T & F & F & T & T & T & T
T & F & T & F & T & T & T & T
T & F & F & F & T & T & T & T
F & T & T & T & T & T & T & T
F & T & F & F & F & T & F & F
F & F & T & F & F & F & T & F
F & F & F & F & F & F & F & F
\hline \end{tabular} Since the truth values in the columns for \( p \lor (q \land r) \) and \( (p \lor q) \land (p \lor r) \) are identical for all possible truth values of \(p, q, \) and \(r\), the given equivalence (the distributive law) is verified.