alleryStart learning →The IMPLICATION $P \to Q$ is FALSE in exactly which case?
AP true, Q true
BP false, Q true
CP true, Q false
DP false, Q false
Answer & Solution
Correct answer: C. P true, Q false
1. The truth table of $P \to Q$:
- T → T: TRUE
- T → F: FALSE (the only false case)
- F → T: TRUE (vacuously)
- F → F: TRUE (vacuously)
2. An implication is FALSE only when the premise holds but the conclusion fails.
3. The 'vacuously true' cases (premise false) sometimes confuse people but are essential — they make implications useful for proving general statements.
_Source: Oscar Levin, "Discrete Mathematics: An Open Introduction", §0.2 (Implications)._
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