Question

If \( p, q \) are true statements and \( r \) is a false statement, then which of the following statements is a true statement?

• A. \( (p \land q) \to r \) is true.
• B. \( (p \to r) \to q \) is false.
• C. \( (p \lor r) \lor q \) is false.
• D. \( (p \leftrightarrow q) \leftrightarrow r \) is false.

Hint:

In logical statements, always evaluate the truth values of individual components before applying the logical operators.

Answer 1

The Correct Option is D

Step 1: Analyze the logical expressions.
We are given that \( p \) and \( q \) are true, and \( r \) is false. Let's evaluate each option using these values: - (A) \( (p \land q) \to r \) is false, because \( p \land q \) is true and \( r \) is false. - (B) \( (p \to r) \to q \) is true, because \( p \to r \) is false, making the implication true regardless of \( q \). - (C) \( (p \lor r) \lor q \) is true, because \( p \) and \( q \) are true. - (D) \( (p \leftrightarrow q) \leftrightarrow r \) is false, because \( p \leftrightarrow q \) is true and \( r \) is false. Thus, the equivalence is false.

Step 2: Conclusion.
Thus, the correct answer is \( (p \leftrightarrow q) \leftrightarrow r \) is false, corresponding to option (D).