Question

Let \( p, q, r \) be three logical statements. Consider the compound statements: \[ S_1: (\neg p \vee q) \vee (\neg p \vee r) \] \[ S_2: p \rightarrow (q \vee r) \] Which of the following is NOT true?

• A. If \( S_2 \) is true, then \( S_1 \) is true
• B. If \( S_2 \) is false, then \( S_1 \) is false
• C. If \( S_2 \) is false, then \( S_1 \) is true
• D. If \( S_1 \) is false, then \( S_2 \) is false

Hint:

Equivalent logical statements will always have the same truth value.

Answer 1

The Correct Option is C

Expanding \( S_1 \): \[ (\neg p \vee q) \vee (\neg p \vee r) \equiv \neg p \vee (q \vee r) \] Expanding \( S_2 \): \[ p \rightarrow (q \vee r) \equiv \neg p \vee (q \vee r) \] Since both \( S_1 \) and \( S_2 \) are equivalent, if \( S_2 \) is false, then \( S_1 \) should also be false, contradicting option (C).