Question

Using truth table, prove that the statement patterns \( p \leftrightarrow q \) and \( (p \land q) \lor (\sim p \land \sim q) \) are logically equivalent.

Hint:

For logical equivalence, compare the truth tables of the two expressions. If all values match, the expressions are equivalent.

Answer 1

Step 1: Construct the truth table.
We need to construct a truth table to show that the two expressions are logically equivalent. Consider the columns for \( p \), \( q \), \( p \leftrightarrow q \), and \( (p \land q) \lor (\sim p \land \sim q) \). \[\begin{array}{|c|c|c|c|c|} \hline p & q & p \leftrightarrow q & (p \land q) \lor (\sim p \land \sim q) \\ \hline T & T & T & T \\ \hline T & F & F & F \\ \hline F & T & F & F \\ \hline F & F & T & T \\ \hline \end{array} \]

Step 2: Explanation.
The columns for \( p \leftrightarrow q \) and \( (p \land q) \lor (\sim p \land \sim q) \) are identical, which shows that the two expressions are logically equivalent.

Final Answer: \[ \boxed{\text{The two expressions are logically equivalent.}} \]