Question

Let P, Q, R and S be statements and suppose that \(P\to Q \to R \to P\); if \(\sim S \to R\), then

• A. \(S \to \sim Q\)
• B. \(\sim Q \to S\)
• C. \(S \to \sim Q\)
• D. \(Q \to \sim S\)

Hint:

Always use contrapositive: \(A\to B\) is equivalent to \(\sim B \to \sim A\).

Answer 1

The Correct Option is B

Step 1: Understand given implication chain.
Given:
\[ P\to Q,\quad Q\to R,\quad R\to P \]
This means all three are logically equivalent:
\[ P \Leftrightarrow Q \Leftrightarrow R \]
Step 2: Use the given statement \(\sim S \to R\).
\[ \sim S \to R \]
Take contrapositive:
\[ \sim R \to S \]
Step 3: Replace \(\sim R\) using equivalence.
Since \(Q \Leftrightarrow R\):
\[ \sim R \Leftrightarrow \sim Q \]
Thus:
\[ \sim Q \to S \]
Final Answer:
\[ \boxed{\sim Q \to S} \]