Question

Which of the following statement patterns is a tautology? \[ S_1 \equiv \neg p \rightarrow (q \leftrightarrow p) \] \[ S_2 \equiv \neg p \vee q \] \[ S_3 \equiv (p \rightarrow q) \land (q \rightarrow p) \] \[ S_4 \equiv (q \rightarrow p) \vee (\neg p \leftrightarrow q) \]

• A. \( S_2 \)
• B. \( S_4 \)
• C. \( S_1 \)
• D. \( S_3 \)

Hint:

To identify tautologies, check if the statement is always true, regardless of the truth values of its components.

Answer 1

The Correct Option is B

Step 1: Understand the statement patterns.
A tautology is a statement that is always true, regardless of the truth values of the individual components. We evaluate each statement: - \( S_1 \equiv \neg p \rightarrow (q \leftrightarrow p) \): This is not a tautology, as its truth depends on the values of \( p \) and \( q \). - \( S_2 \equiv \neg p \vee q \): This is not a tautology, as it depends on the values of \( p \) and \( q \). - \( S_3 \equiv (p \rightarrow q) \land (q \rightarrow p) \): This is not a tautology, as it can be false for certain values of \( p \) and \( q \). - \( S_4 \equiv (q \rightarrow p) \vee (\neg p \leftrightarrow q) \): This is a tautology because it is true regardless of the truth values of \( p \) and \( q \).
Step 2: Conclusion.
Thus, the correct answer is option (B), \( S_4 \).