In mathematics, one of the most famous ways of proving theorems and propositions is to prove by contradiction. Actually, this is a logical way of thinking based on "the law on non-contradiction (LNC)" as first formalized as a metaphysical principle by Aristotle.

This way of proving says that a statement P and its negation (non)P can not happen at the same time. For example, day and night can not happen at the same time; similarly, left and right, up and down, and full and empty.

So, to prove a theorem or a proposition, you may either prove that its statement is right, or the negation of its statement is wrong.

Using the contradiction in proving a theorem or a proposition in mathematics starts usually by assuming that the statement of this theorem or proposition is wrong. Then, the researcher should find a mistake as a result of his assumption. Thus, the statement must not be wrong by contradiction, and so it is true.

For example, we will prove, using contradiction, the following statement P: "If `x^2`

is even, then `x`

is even". Suppose to get a contradiction that `x`

is odd, that is `x=n+1`

, where `n`

is an even number. So, `x^2=(n+1)^2=n^2+2n+1`

. Then `x^2-(n)^2-2n=1`

an odd number, hence the number `y= x^2-(n)^2-2n`

is an odd number. On the other hand, `x^2`

is even, `(n)^2`

is even since `n`

is even and `2n`

is even, which implies that `y`

is even. So, `y`

is odd and even at the same time, which is a contradiction. Therefore, the statement P is true.

### Conclusion

Although there are a lot of ways for proving theorems and propositions in mathematics, proving using contradiction stays the more famous and familiar way mathematicians like to use.