Proof By Contraposition

Discrete Math: A Proof By Contraposition

Proof by contraposition is a type of proof used in mathematics and is a rule of inference. In logic the contrapositive of a statement can be formed by reversing the direction of inference and negating both terms for example :

p → q
= -p ← -q
= -q → -p

This simply means “if p, then q” is drawn from the single premise “if not q, then not p.” A statement and its contrapositive are logically equivalent: if the statement is true, then its contrapositive is true, and vice versa.

Let’s prove or show that n to the power of 2 is a even number using contraposition.

Prove if n² is even then n is even.

Determine our p and q value:
p → q = “if n² is even then n is even”

p: “n² is even”
q: “n is even”

p: “n² is NOT even” = “n² is odd”
-q: “n is NOT even” = “n is odd”

NOTE: If a number is not even then it is odd

This means our contrapositive is :
-q → -p = “if n is odd then n² is odd”