Question:
If \( p \to (\sim p \lor q) \) is false, then the truth values of \( p \) and \( q \) are, respectively
If \( p \to (\sim p \lor q) \) is false, then the truth values of \( p \) and \( q \) are, respectively
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An implication is false only when antecedent is true and consequent is false. Use this to work backwards.
Updated On: Jun 4, 2026
- T, F
- F, F
- F, T
- T, T
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
We need truth values of \(p\) and \(q\) such that the compound statement \(p \to (\sim p \lor q)\) is false.
Step 2: Key Formula or Approach:
Implication \(X \to Y\) is false only when \(X\) is true and \(Y\) is false.
Step 3: Detailed Explanation:
Let \(X = p\) and \(Y = (\sim p \lor q)\). For \(p \to (\sim p \lor q)\) to be false, we require \(p = \text{True}\) and \((\sim p \lor q) = \text{False}\).
If \(p = \text{True}\), then \(\sim p = \text{False}\). So \(\sim p \lor q = \text{False} \lor q = q\). For this to be false, \(q\) must be False.
Thus \(p = \text{True}\), \(q = \text{False}\).
Step 4: Final Answer:
Option (A) is correct.
We need truth values of \(p\) and \(q\) such that the compound statement \(p \to (\sim p \lor q)\) is false.
Step 2: Key Formula or Approach:
Implication \(X \to Y\) is false only when \(X\) is true and \(Y\) is false.
Step 3: Detailed Explanation:
Let \(X = p\) and \(Y = (\sim p \lor q)\). For \(p \to (\sim p \lor q)\) to be false, we require \(p = \text{True}\) and \((\sim p \lor q) = \text{False}\).
If \(p = \text{True}\), then \(\sim p = \text{False}\). So \(\sim p \lor q = \text{False} \lor q = q\). For this to be false, \(q\) must be False.
Thus \(p = \text{True}\), \(q = \text{False}\).
Step 4: Final Answer:
Option (A) is correct.
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