2. Proof by Contrapositive

Video coming soon2 worked examples

Theory

To prove “if $P$ then $Q$ ”, it is sometimes easier to prove the logically equivalent contrapositive: “if not $Q$ then not $P$ ”. The two statements are always true or false together.

(P \Rightarrow Q) \quad\equiv\quad (\lnot Q \Rightarrow \lnot P)

The Golden Rule: the contrapositive negates and swaps the two parts. It is not the converse “if $Q$ then $P$ ”, which is not equivalent.

⚠️ Common Examiner Traps

Contrapositive vs converse: the converse $Q \Rightarrow P$ is a different statement — do not prove that instead.
Negating carelessly: the negation of “even” is “odd”; make sure the negations are exact.
Stopping early: once the contrapositive is proved, state that the original statement therefore holds.

Worked examples

Example 1

Prove, by contrapositive, that for any integer $n$ , if $n^2$ is even then $n$ is even.

Step 1: The contrapositive is: “if $n$ is odd then $n^2$ is odd”. Assume $n$ is odd, so $n = 2k+1$ .

Step 2: Square it:

n^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1

Step 3: This is of the form $2 \times \text{integer} + 1$ , so $n^2$ is odd. The contrapositive holds, so the original statement is true.

Example 2

Prove, by contrapositive, that for any integer $n$ , if $3n + 2$ is odd then $n$ is odd.

Step 1: The contrapositive is: “if $n$ is even then $3n + 2$ is even”. Assume $n$ is even, so $n = 2k$ .

Step 2: Substitute:

3n + 2 = 3(2k) + 2 = 6k + 2 = 2(3k + 1)

Step 3: This is even. The contrapositive holds, so the original statement is true.