c. Let p: I am in California. Let q: I am in Napa Valley. Given the statement "If I am not in California, then I am not in Napa Valley." The negation of the conditional statement is \[ \sim(\sim q \rightarrow \sim p) \] \( O \sim(\sim p \rightarrow \sim q) \) \( O \sim(p \rightarrow \sim q) \) \( O \sim(\sim p \rightarrow q) \)

To find the negation of the conditional statement "If I am not in California, then I am not in Napa Valley," we start by identifying the components of the statement. 1. **Identify the components:** - Let \( p \): I am in California. - Let \( q \): I am in Napa Valley. - The original statement can be expressed as: \[ \sim p \rightarrow \sim q \] - This reads as "If I am not in California, then I am not in Napa Valley." 2. **Negate the conditional statement:** - The negation of a conditional statement \( A \rightarrow B \) is given by \( A \land \sim B \). - Therefore, the negation of \( \sim p \rightarrow \sim q \) is: \[ \sim(\sim p \rightarrow \sim q) = \sim(\sim p) \land \sim(\sim q) = p \land q \] - This means "I am in California and I am in Napa Valley." 3. **Evaluate the options:** - \( O \sim(\sim q \rightarrow \sim p) \): This is not the negation we derived. - \( O \sim(\sim p \rightarrow \sim q) \): This is the negation we derived, which simplifies to \( p \land q \). - \( O \sim(p \rightarrow \sim q) \): This is not the negation we derived. - \( O \sim(\sim p \rightarrow q) \): This is not the negation we derived. Thus, the correct negation of the conditional statement is: \[ O \sim(\sim p \rightarrow \sim q) \]