c. Let p : You earn a bachelor's degree. Let q : Your income potential increase. Given the statement "If you earn a bachelor's degree, then your income potential increases." The negation of the conditional statement is \( \left.\qquad \begin{array}{l}O(p \rightarrow q) \\ O \\ \sim(p \rightarrow \sim q) \\ O(\sim p \rightarrow \sim q) \\ \end{array}\right) \sim(\sim p \rightarrow q) \)

To find the negation of the conditional statement "If you earn a bachelor's degree, then your income potential increases," we start by identifying the components of the statement: - Let \( p \): You earn a bachelor's degree. - Let \( q \): Your income potential increases. The original conditional statement can be expressed in logical terms as: \[ p \rightarrow q \] This means "if \( p \) is true, then \( q \) is also true." ### Step 1: Understanding the Negation of a Conditional Statement The negation of a conditional statement \( p \rightarrow q \) is given by: \[ \sim(p \rightarrow q) \equiv p \land \sim q \] This means that the negation is true when \( p \) is true and \( q \) is false. In other words, you earn a bachelor's degree, but your income potential does not increase. ### Step 2: Analyzing the Options Now, let's analyze the options provided: 1. \( O(p \rightarrow q) \) - This is the original statement, not its negation. 2. \( O \) - This option is unclear and does not represent a logical statement. 3. \( \sim(p \rightarrow \sim q) \) - This is not the correct negation. 4. \( O(\sim p \rightarrow \sim q) \) - This represents a different logical relationship and is not the negation of the original statement. 5. \( \sim(\sim p \rightarrow q) \) - This is also not the correct negation. ### Conclusion The correct negation of the statement "If you earn a bachelor's degree, then your income potential increases" is: \[ p \land \sim q \] This means that you earn a bachelor's degree, but your income potential does not increase. None of the provided options correctly represent this negation.