To prove a mathematical statement, you must show it is a tautology. One way to show a statement is a tautology is to show that the negation of the statement is a contradiction.

A proof by contradiction is an argument that shows the negation of a statement is a contradiction.

It may seem unnecessary to have such a thing as proof by contradiction, but it sometimes happens that it is easier to prove the negation of a statement is a contradiction than to prove a statement is a tautology. Proof by contradiciton is a strategy, and it takes practice to situations in which it makes an argument easier.

Example

Claim: If \(x\) is rational and \(y\) is irrational, then \(x + y\) is irrational.

Proof: Remember that a rational number is a number of the form \(\frac{a}{b}\) where \(a\) and \(b\) are integers. An irrational number is a real number that cannot be written as \(\frac{a}{b}\) for some integers \(a\) and \(b.\) Examples of irrationals include \(\sqrt{2}\) and \(\pi.\)

It is hard to argue that the claim is a tautology, but consider the negation of the claim: "There exists a rational number \(x\) and an irrational number \(y\) such that \(x+y\) is rational."

We can prove the claim by showing that the negation is a contradiction. If \(x\) is rational, then there exists integers \(a\) and \(b\) such that \(x = \frac{a}{b}.\) Also, if \(x+y\) is rational, then there exists integers \(c\) and \(d\) such that \(x+y = \frac{c}{d}.\) Then \begin{align} & x + y = \frac{c}{d} \Rightarrow \\ & \frac{a}{b} + y = \frac{c}{d} \Rightarrow \\ & y = \frac{c}{d} - \frac{a}{b} \\ & y = \frac{bc-ad}{bd} \end{align} This shows that \(y\) can be written as a rational, but this is a contradiciton to \(y\) being irrational. Since this is a contradiction, the original claim must be true.