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: Work backwards
A false conditional forces the antecedent to be T and the consequent F, so $p \wedge q = F$ and each of the three conjuncts is T.

Step 2: Deduce
$\sim r = T$ gives $r = F$; $q \to r = T$ with $r = F$ forces $q = F$; $p \vee q = T$ with $q = F$ forces $p = T$.

Step 3: Test each claim
$p \to q$: F (claim true). $p \to r$: F (claim true). $\sim p \wedge r$: F (claim true). $\sim q \to p$: T, but the option says it is False, so that claim is wrong.

Step 4: Answer
Option (C) is the one that is NOT true.

Final Answer:
Option (C) is not true. This is option (C). \[ \boxed{\text{(C) }(\sim q)\to p \text{ is not False}} \]