Question

P, Q, R and S are four statements. Relation between these statements is as follows. I. If P is true, then Q must be true.
II. If Q is true, then R must be true.
III. If S is true, then either Q is false or R is false.
Which of the following must be true?

• A. If P is true, then S is false.
• B. If S is false, then Q must be true.
• C. If Q is true, then P must be true.
• D. If R is true, then Q must be true.

Hint:

When working with logical statements, carefully consider the consequences of each given relationship, and check for contradictions or confirmations from other statements.

Answer 1

The Correct Option is A

We are given the following relations: - I. If \( P \) is true, then \( Q \) must be true. - II. If \( Q \) is true, then \( R \) must be true. - III. If \( S \) is true, then either \( Q \) is false or \( R \) is false. We need to determine which of the given options must be true. Step 1: Analyze the implications - From I: If \( P \) is true, \( Q \) must be true. - From II: If \( Q \) is true, \( R \) must be true. This means that if \( P \) is true, then both \( Q \) and \( R \) are true. Step 2: Analyze the condition in III From III: If \( S \) is true, then either \( Q \) is false or \( R \) is false. - But, from Step 1, we know that if \( P \) is true, then \( Q \) and \( R \) are both true. - Therefore, if \( P \) is true, \( S \) must be false, because \( S \) cannot be true when both \( Q \) and \( R \) are true. Thus, the correct answer is (A) If \( P \) is true, then \( S \) is false.