A statement is equivalent to its contrapositive. So, one way to prove a statement is to prove its contrapositive.
A proof by contrapositive is an argument that shows the contrapositive of a statement is a tautology.
It may seem unnecessary to have such a thing as proof by contrapositive, but it sometimes happens that it is easier to prove the contrapositive of a statement than to prove a given statement. It takes practice to recognize situations in which proof by contrapositive makes an argument easier.
Claim: If \(x\) and \(y\) are integers such that \(x+y\) is odd, then at least one of \(x\) or \(y\) is odd.
Proof: The contrapositive of the given statement is "If \(x\) and \(y\) are integers such that neither \(x\) nor \(y\) is odd, then \(x+y\) is not odd." Since "not odd" means "even," we can make the claim easier to read by writing it as "If \(x\) and \(y\) are even integers, then \(x+y\) is even."
If \(x\) and \(y\) are even, then by definition of even there exists integers \(a\) and \(b\) such that \(x = 2a\) and \(y = 2b.\) So,
\[x+y = 2a + 2b = 2(a+b)\]
So, \(x+y\) is \(2\) times an integer. Therefore, \(x+y\) is even. This proves the contrapositive, which equivalently proves the claim.