Which statement correctly expresses the logical negation of 'Robert has black hair and Sally does not have black hair'?

Which statement correctly expresses the logical negation of 'Robert has black hair and Sally does not have black hair'?

• A. Robert has black hair or Sally has black hair.
• B. Robert does not have black hair and Sally has black hair.
• C. Robert does not have black hair or Sally has black hair.
• D. Robert has black hair and Sally does not have black hair.

Answer: (C)

Negating a statement that combines two claims with “and” becomes saying that at least one of those claims is not true. If we call the first claim “Robert has black hair” and the second claim “Sally does not have black hair,” the original statement is a conjunction of those two. Its negation is “not (P and Q),” which equals “not P or not Q.” Here, not P is “Robert does not have black hair,” and not Q is “Sally has black hair.” So the correct way to express the negation is: “Robert does not have black hair or Sally has black hair.” This means at least one part of the original claim is false.

Negating a statement that combines two claims with “and” becomes saying that at least one of those claims is not true. If we call the first claim “Robert has black hair” and the second claim “Sally does not have black hair,” the original statement is a conjunction of those two. Its negation is “not (P and Q),” which equals “not P or not Q.” Here, not P is “Robert does not have black hair,” and not Q is “Sally has black hair.” So the correct way to express the negation is: “Robert does not have black hair or Sally has black hair.” This means at least one part of the original claim is false.