Questions: Proving by Contrapositive

Question 1

To prove 'If n is even, then n² is even,' a student instead proves 'If n² is odd, then n is odd.' Has she proven the original statement?

• A. No — she has proven the converse, which is not logically equivalent to the original
• B. Yes — she has proven the contrapositive, which is logically equivalent to the original
• C. No — she must prove both the statement and its contrapositive to establish the original
• D. Yes — she has proven the inverse, which is equivalent to the original

Answer: (B)

She has proven 'If n² is odd, then n is odd,' which is the contrapositive of the original. The contrapositive of P → Q is ¬Q → ¬P — here, negating 'n² is even' gives 'n² is odd' and negating 'n is even' gives 'n is odd.' Since the contrapositive is logically equivalent to the original (same truth table), proving one proves the other. You do not need to prove both.

Question 2

A student needs to prove 'If 3n + 2 is odd, then n is odd.' She finds the direct approach awkward. She writes the contrapositive as 'If n is odd, then 3n + 2 is odd.' Is this correct, and why or why not?

• A. Yes — she swapped the two parts of the conditional, which is the contrapositive
• B. No — she wrote the converse (swapping without negating); the correct contrapositive is 'If n is even, then 3n + 2 is even'
• C. Yes — the contrapositive just requires negating the conclusion, which she did
• D. No — the contrapositive requires negating only the hypothesis, giving 'If n is not odd, then 3n + 2 is odd'

Answer: (B)

She wrote the converse (Q → P), not the contrapositive (¬Q → ¬P). The contrapositive of 'If 3n+2 is odd, then n is odd' requires negating both parts and swapping: 'If n is even (¬'n is odd'), then 3n+2 is even (¬'3n+2 is odd').' This version is easy to prove directly: n = 2k gives 3n+2 = 6k+2 = 2(3k+1), which is even. The converse is not equivalent to the original and proves nothing about it.

Question 3

Proving the contrapositive ¬Q → ¬P is a valid way to establish P → Q because the two conditionals are logically equivalent — they have the same truth table.

• T. True
• F. False

Answer: True

This is exact, not approximate. P → Q and ¬Q → ¬P are both false only when P is true and Q is false. In all other cases both are true. This complete equivalence is the entire justification for contrapositive proof — you are not approximating or using a shortcut; you are proving the same logical claim in an equivalent form.

Question 4

Proving the converse (Q → P) of a statement is a valid way to establish the original statement (P → Q), just like proving the contrapositive.

• T. True
• F. False

Answer: False

The converse Q → P is NOT logically equivalent to P → Q. They can have different truth values: 'If it is a square, then it is a rectangle' is true, but its converse 'If it is a rectangle, then it is a square' is false. By contrast, the contrapositive ¬Q → ¬P is always logically equivalent to P → Q. The converse and contrapositive look superficially similar — both swap the two parts — but the contrapositive negates both parts, and that negation is everything.

Question 5

Why would a mathematician choose to prove a statement by contrapositive rather than directly, and what makes this choice logically valid?

Model answer: A mathematician chooses the contrapositive when assuming ¬Q (the negation of the conclusion) is more useful than assuming P (the original hypothesis). For example, proving 'if n² is even, then n is even' is awkward directly, but the contrapositive 'if n is odd, then n² is odd' immediately gives n = 2k+1, which is easy to square. The choice is valid because the contrapositive is logically equivalent to the original — proving one proves the other without any loss.

The strategic question is always: which hypothesis gives me more to work with? If the conclusion's negation (¬Q) leads more naturally to the hypothesis's negation (¬P) than P leads to Q, contrapositive proof is the cleaner path. The key check before starting: make sure you have written ¬Q → ¬P and not the converse Q → P. Both swap the parts, but only the contrapositive negates them.