Answer to Question #269603 in Discrete Mathematics for maven

Question

I. Create a truth table for all of the 11 Logical Equivalence.

II. Find if the following are logically equivalent

   1.(~p v q) ^ (~q) <=> ~(p v q)

   2.(p ^ ~q) v (~p v q) <=> T

Answer 1

Let us create a truth table for all of the following formulas and find if the following are logically equivalent.

1. $f(p,q)=(\sim p \lor q) \land (\sim q) \leftrightarrow \sim(p \lor q)$

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

Since the formula $f(p,q)$ is tautology, we conclude that the formulas $(\sim p \lor q) \land (\sim q)$ and $\sim(p \lor q)$ are logically equivalent.

2. $f(p,q)=(p \land \sim q) \lor (\sim p \lor q) \leftrightarrow T$

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

Since the formula $f(p,q)$ is tautology, we conclude that the formulas $(p \land \sim q) \lor (\sim p \lor q)$ and $T$ are logically equivalent.