Question

Show, by the use of the truth table/matrix, that the statement (p v q) v [( ¬p) ∧ (¬q)] is a tautology.

Answer 1

Let us construct the truth table of the statement $(p\lor q) \lor [( ¬p) ∧ (¬q)]$

$\ \begin{array}{||c|c||c|c|c|c|c||} \hline\hline p & q & \neg p & \neg q & p\lor q&( ¬p) ∧ (¬q) & (p\lor q) \lor [( ¬p) ∧ (¬q)]\\ \hline\hline 0 & 0 & 1 & 1 & 0 & 1 & 1\\ \hline 0 & 1 & 1 & 0 & 1 &0 & 1 \\ \hline 1 & 0 & 0 & 1 & 1 & 0 & 1 \\ \hline 1 & 1 & 0 & 0 & 1 & 0 & 1 \\ \hline\hline \end{array}$

It follows that the statement $(p\lor q) \lor [( ¬p) ∧ (¬q)]$ is a tautology.