Question

The truth values of \( p, q \) and \( r \) for which \( (p \wedge q) \vee (\sim r) \) has truth value F are respectively:

• A. F, T, F
• B. F, F, F
• C. T, T, T
• D. T, F, F
• E. F, F, T

Hint:

Always start with the "OR" (\( \vee \)) gate when it is False, because it creates a very strict requirement: everything must be False.

Answer 1

Answer: (E)

Concept: For a disjunction (\( \vee \)) to be False (F), both components of the disjunction must be False. For a conjunction (\( \wedge \)) to be False, at least one of its components must be False.

Step 1: Break down the main operator.
The expression is \( (p \wedge q) \vee (\sim r) \). Since the result is \( F \), we must have: \[ (p \wedge q) = F \quad \text{AND} \quad (\sim r) = F \]

Step 2: Solve for \( r \).
If \( \sim r = F \), then \( r \) must be True (T).

Step 3: Solve for \( p \) and \( q \).
We need \( (p \wedge q) = F \). This happens if \( p \) is F, or \( q \) is F, or both are F. Looking at the options where \( r = T \): Option (E) gives \( p = F, q = F, r = T \). Let's check: \( (F \wedge F) \vee (\sim T) \equiv F \vee F \equiv F \). This satisfies the condition.