Question

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

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

Hint:

A contradiction is a statement that is false for all possible truth values.

Answer 1

The Correct Option is B

Step 1: Analyze \(S_1\).
\[ S_1 = (p \to q) \land (p \land \sim q) \] Now, \[ p \to q \equiv (\sim p \lor q) \] So \[ S_1 = (\sim p \lor q) \land p \land \sim q \]
Step 2: Simplify the expression.
The statement requires \(p\) to be true and \(q\) to be false, while simultaneously demanding \((\sim p \lor q)\), which becomes false under these conditions.

Step 3: Conclusion.
Thus, \(S_1\) is always false for all truth values of \(p\) and \(q\). Hence, it is a contradiction.