Question

Which of the following statement patterns is a tautology?
\[ \begin{aligned} S_1 &: (\sim q \land p) \land q
S_2 &: [p \land (p \rightarrow q)] \rightarrow q
S_3 &: (p \land q) \land (\sim p \lor \sim q)
S_4 &: (p \land q) \rightarrow r \end{aligned} \]

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

Hint:

A tautology is a statement that is true for all possible truth values—check implications carefully.

Answer 1

The Correct Option is D

Step 1: Analyze each statement.
\( S_1 \): Contains both \( q \) and \( \sim q \), hence it is always false (contradiction).
\( S_3 \): \( (p \land q) \land (\sim p \lor \sim q) \) cannot be true simultaneously, hence not a tautology.
\( S_4 \): \( (p \land q) \rightarrow r \) depends on the truth value of \( r \), so it is not always true.
\( S_2 \): If \( p \land (p \rightarrow q) \) is true, then \( q \) must be true. Hence the implication is always true.

Step 2: Conclusion.
The statement pattern which is always true is \( S_2 \).