Question

If a statement $q$ has truth value False and $(p \land q) \leftrightarrow r$ has truth value True then which of the following has truth value true?

• A. $p \land q$
• B. $p \lor r$
• C. $p \land r$
• D. $(p \land r) \rightarrow (p \lor r)$

Hint:

"False implies anything" ($F \rightarrow \text{Anything}$) is always True. This is a common pattern in logic questions.

Answer 1

The Correct Option is D

Step 1: Concept
Analyze the truth values of the components based on the logical connectives $\land$ (AND), $\leftrightarrow$ (Biconditional), and $\rightarrow$ (Implication).

Step 2: Meaning
If $q$ is False, then $(p \land q)$ is always False regardless of $p$.

Step 3: Analysis
Given $(p \land q) \leftrightarrow r$ is True and $(p \land q)$ is False, $r$ must be False (as $F \leftrightarrow F$ is True). Now check options: (A) $p \land F$ is False. (B) $p \lor F$ depends on $p$. (C) $p \land F$ is False. (D) $(p \land F) \rightarrow (p \lor F) \implies F \rightarrow p$. An implication with a False antecedent is always True.

Step 4: Conclusion
Option (D) is a tautology in this context and is therefore True. Final Answer: (D)