d. Let \( p: \) I build a rocket. Let \( \mathbf{q}: \) I fly to Mars. Given the statement "If I build a rocket, then I will not fly to Mars." The logically equivalent negation of the conditional statement is \[ p \wedge \sim q \] \( O \sim p \wedge q \) \( O \sim p \wedge \sim q \) \( O p \wedge q \)

We start with the conditional statement: \[ p \to \sim q \] This means "If I build a rocket, then I will not fly to Mars." The negation of the conditional is: \[ \sim (p \to \sim q) \] Recall that a conditional \( p \to \sim q \) is logically equivalent to \( \sim p \lor \sim q \). Thus, we have: \[ \sim (p \to \sim q) = \sim (\sim p \lor \sim q) \] Using De Morgan's Law, which states that \( \sim (A \lor B) \equiv \sim A \wedge \sim B \), we obtain: \[ \sim (\sim p \lor \sim q) = \sim (\sim p) \wedge \sim (\sim q) \] Since \( \sim (\sim p) \) is equivalent to \( p \) and \( \sim (\sim q) \) is equivalent to \( q \), the expression simplifies to: \[ p \wedge q \] Thus, the logically equivalent negation of the conditional statement is: \[ p \wedge q \]