Question

Negation of the statement \( (p \land r) \rightarrow (r \lor q) \) is:

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

Hint:

To negate a conditional statement of the form \( A \rightarrow B \), use the equivalence \( \sim (A \rightarrow B) \equiv A \land \sim B \).

Answer 1

The Correct Option is D

The negation of the statement \( (p \land r) \rightarrow (r \lor q) \) is derived as follows:\[\sim \left[ (p \land r) \rightarrow (r \lor q) \right]\]Applying the logical equivalence \( A \rightarrow B \equiv \sim A \lor B \), the expression becomes:\[\sim \left[ \sim (p \land r) \lor (r \lor q) \right]\]Further simplification yields:\[(p \land r) \land (\sim r \land \sim q)\]Therefore, the correct option is \(\boxed{(d)}\).