Question

If the truth value of \([(p∨q)∧(q\rightarrow r)∧(\sim r)]\rightarrow (p∧q)\) is false, then which of the following is NOT true?

• A. Truth value of \(p\rightarrow q\) is False
• B. Truth value of \(p\rightarrow r\) is False.
• C. Truth value of \((\sim q)\rightarrow p\) is False.
• D. The truth value of \((\sim p)∧r\) is False.

Hint:

A conditional is false only when the antecedent is true and the consequent is false.

Answer 1

The Correct Option is C

Step 1: Understand the concept
\(A \to B\) is false only when \(A\) is true and \(B\) is false. The given statement is false, so \((p \vee q) \wedge (q \to r) \wedge (\sim r)\) is true while \(p \wedge q\) is false.

Step 2: Find the truth values
\(\sim r\) true gives \(r = F\). Then \(q \to r\) true needs \(q = F\). Then \(p \vee q\) true needs \(p = T\). Check: \(p \wedge q = F\), as needed. So \((p, q, r) = (T, F, F)\).

Step 3: Evaluate each option
(A) \(p \to q\) = \(T \to F\) = False, so "is False" is true. (B) \(p \to r\) = \(T \to F\) = False, so "is False" is true. (D) \((\sim p) \wedge r\) = \(F \wedge F\) = False, so "is False" is true.

Step 4: Option (C)
\((\sim q) \to p\) = \(T \to T\) = True. So the claim that its truth value is False is not true. Option (C) is the answer.

Final Answer:
Only option (C) makes a wrong claim. This is option (C). \[ \boxed{\text{(C) }(\sim q)\to p \text{ is not False}} \]