Question

Suppose that p and q are any statements. By constructing the truth tables, show that the

statement ¬ (p V q) & (¬ p) ∧ (¬ q) are logically equivalent.

Answer 1

Let us construct the trush table for the formulas $¬ (p \lor q)$ and $(¬ p) \land (¬ q) :$

$\begin{array}{||c|c||c|c|c|c|c||} \hline\hline p & q & p \lor q & ¬ p & ¬ q & ¬ (p \lor q) & (¬ p) \land (¬ q) \\ \hline\hline 0 & 0 & 0 & 1 & 1 & 1 & 1\\ \hline 0 & 1 & 1 & 1 & 0 & 0 & 0\\ \hline 1 & 0 & 1 & 0 & 1 & 0 & 0\\ \hline 1 & 1 & 1 & 0 & 0 & 0 & 0\\ \hline\hline \end{array}$

Since the last two columns are coinside, the formulas $¬ (p \lor q)$ and $(¬ p) \land (¬ q)$ are logically equivalent.