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).

Answer 2

Step 1: Simplifying \( S_1 \)
The compound statement \( S_1 \) is: \[ S_1: (\neg p \vee q) \vee (\neg p \vee r) \] Using the associative and commutative properties of logical OR (\( \vee \)), we can simplify \( S_1 \): \[ S_1 = \neg p \vee (q \vee r) \] Thus, \( S_1 \) simplifies to: \[ S_1 = \neg p \vee (q \vee r) \] This means that \( S_1 \) is true if either \( p \) is false or at least one of \( q \) or \( r \) is true.

Step 2: Simplifying \( S_2 \)
The compound statement \( S_2 \) is: \[ S_2: p \rightarrow (q \vee r) \] Recall that the implication \( p \rightarrow (q \vee r) \) is equivalent to: \[ S_2 = \neg p \vee (q \vee r) \] Thus, \( S_2 \) is true if either \( p \) is false or at least one of \( q \) or \( r \) is true.

Step 3: Analyzing the truth values of \( S_1 \) and \( S_2 \)
- \( S_1 \) and \( S_2 \) are both equivalent in their form: \[ S_1 = \neg p \vee (q \vee r) \] \[ S_2 = \neg p \vee (q \vee r) \] Therefore, \( S_1 \) and \( S_2 \) are logically equivalent, meaning they will have the same truth value.

Step 4: Evaluating the given statement
The question asks which of the following is NOT true: "If \( S_2 \) is false, then \( S_1 \) is true." Since \( S_1 \) and \( S_2 \) are logically equivalent, if \( S_2 \) is false, then \( S_1 \) will also be false. Therefore, the statement "If \( S_2 \) is false, then \( S_1 \) is true" is NOT true.

Step 5: Final Answer
The correct answer is:

If \( S_2 \) is false, then \( S_1 \) is true