Question

If the truth value of the Boolean expression \(((p \vee q) \wedge (q \to r) \wedge (\sim r)) \to (p \wedge q)\) is false, then the truth values of the statements \(p, q, r\) respectively can be :

• A. \(T, F, T\)
• B. \(T, F, F\)
• C. \(F, T, F\)
• D. \(T, T, F\)

Hint:

For questions on "False" implications, always start from the end. Set the result \(Y\) to False and the start \(X\) to True, then deduce individual statement values step-by-step.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The logical implication \(X \to Y\) is false only in one scenario: when the antecedent \(X\) is True and the consequent \(Y\) is False.
Step 2: Detailed Explanation:
Let \(X = ((p \vee q) \wedge (q \to r) \wedge (\sim r))\) and \(Y = (p \wedge q)\). For \(X \to Y\) to be False, we need \(X = T\) and \(Y = F\). For \(X = (p \vee q) \wedge (q \to r) \wedge (\sim r)\) to be True, all three parts must be True: 1. \(p \vee q = T\) 2. \(q \to r = T\) 3. \(\sim r = T \implies r = F\) Since \(r = F\), for \(q \to r\) to be True, \(q\) must be False (\(F \to F = T\)). Now check \(p \vee q = T\): since \(q = F\), we must have \(p = T\). Now check the consequent \(Y = p \wedge q\): \(Y = T \wedge F = F\). This matches our required condition for the implication to be false. So, \(p = T, q = F, r = F\).
Step 3: Final Answer:
The values are \(T, F, F\).