Consider the following statements: m = Edna is respectful n = Edna is brilliant, If m ⇒ n, which of the following is valid?

Analyze the question: We are given the conditional statement m ⇒ n ("If Edna is respectful, then Edna is brilliant"). We must pick the option that is a valid statement that follows from it.

Concept: For a conditional p ⇒ q, there are three related statements:

• Converse: q ⇒ p (not logically equivalent)
• Inverse: ¬p ⇒ ¬q (not logically equivalent)
• Contrapositive: ¬q ⇒ ¬p (logically equivalent to the original)

Solution:

1. Original: m ⇒ n
2. Contrapositive: ¬n ⇒ ¬m, which means "If Edna is not brilliant, then Edna is not respectful."

Checking each option:

• A. ¬n ⇒ ¬m: this is the contrapositive, so it is always valid.
• B. ¬m ⇒ ¬n: this is the inverse, which is not necessarily valid. Edna could be brilliant without being respectful.
• C. n ⇒ ¬m: this does not follow from m ⇒ n.
• D. m ⇒ n: this is the given statement itself. It is true by assumption, but it is not a new deduction. The question asks which is valid, and the intended answer is the equivalent form, the contrapositive.

A note on D: D is technically true because it was given.

Answer: A

Tip: Remember "flip and negate." To get the contrapositive, swap the two statements and negate both. The contrapositive always has the same truth value as the original. The converse and inverse do not.