Proof by Contrapositive
Starting with an implication
e.g. If triangle ABC is right angled then 𝑐 2 = 𝑎2 + 𝑏 2 .
The contrapositive is formed by negating both statements and reversing the
direction of the implication
2
2
e.g. If 𝑐 ≠ 𝑎 + 𝑏 2 then the triangle is not right angled.
Statement: P => Q
Contrapositive: ~Q => ~P
If P => Q is true, then the contrapositive is also true.
If P => Q is false, then the contrapositive is also false.
Example
Prove that if n3 is odd then n is odd.
Contrapositive: If n is even then n 3 is even.
n = 2k
n 3 = (2k) 3
n 3 = 8k 3
n 3 = 2(4k 3)
=> n 3 is even
The contrapositive is true therefore the original statement is true.