Question

Let p, q, r be three logical statements. Consider the compound statements
\(S_{1}:[(p\rightarrow q)\wedge \sim q]\rightarrow r\) and
\(S_{2}:(p\vee q)\vee r\)
Then 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 \(S_{1}\) is False, then \(S_{2}\) is False.
• D. If \(S_{2}\) is False, then \(S_{1}\) is True.

Hint:

Simplify the first statement using the equivalence of an implication with a disjunction and compare it with the second.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Use \((p\rightarrow q)\equiv (\sim p\vee q)\) and \(a\rightarrow b\equiv \sim a\vee b\).

Step 2: Simplify \(S_{1}\)
\((p\rightarrow q)\wedge \sim q\equiv (\sim p\vee q)\wedge \sim q\equiv \sim p\wedge \sim q\).
So \(S_{1}\equiv (\sim p\wedge \sim q)\rightarrow r\equiv (p\vee q)\vee r\).

Step 3: Compare with \(S_{2}\)
We get \(S_{1}\equiv (p\vee q)\vee r=S_{2}\), so \(S_{1}\) and \(S_{2}\) always have the same truth value.

Step 4: Check the Options
Options (A), (B) and (C) say the truth values are equal, which is correct.
Option (D) says that when \(S_{2}\) is False, \(S_{1}\) is True, which is impossible.

Final Answer:
Option (D) is not true because \(S_{1}\) and \(S_{2}\) are equivalent. \[ \boxed{\text{(D) }\text{If } S_{2} \text{ is False, then } S_{1} \text{ is True.}} \]