Question

If \( \{ (p \land \sim q) \land (p \land r) \} \rightarrow (p \lor q) \) has truth value false, then truth values of the statements \( p, q, r \) are respectively

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

Hint:

In logical expressions, for an implication to be false, the antecedent (left side) must be true, and the consequent (right side) must be false. Use this to simplify logical equations.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
An implication \( \text{A} \to \text{B} \) is false ONLY if \( \text{A} \) is true and \( \text{B} \) is false.
Step 2: Detailed Explanation:
Given: \( [\dots] \to (\sim p \lor q) \) is False.
1. Consequent is False: \( \sim p \lor q \equiv \text{F} \).
This means \( \sim p \) is F and \( q \) is F.
\( \sim p \equiv \text{F} \Rightarrow p \equiv \text{T} \).
So, \( p = \text{T} \) and \( q = \text{F} \).
2. Antecedent is True: \( (p \land \sim q) \land (p \land r) \equiv \text{T} \).
For a conjunction to be true, all parts must be true.
\( p \land \sim q \equiv \text{T} \Rightarrow \text{T} \land \sim \text{F} \equiv \text{T} \) (Matches).
\( p \land r \equiv \text{T} \Rightarrow \text{T} \land r \equiv \text{T} \Rightarrow r \equiv \text{T} \).
So, \( p = \text{T}, q = \text{F}, r = \text{T} \).
Step 3: Final Answer:
The truth values are T, F, T.