We have had several examples on proof by contradiction in this blog like irrationality of **square root of 3 **and the **sum of root of 2 and root of 3**. In this post, we will have another detailed example.

In proof by contradiction, we want to change the statement “if P then Q” to “if NOT Q then NOT P.” These two statements are equivalent, so if we can prove the latter, then we have proved the former. Consider the following theorem.

**Theorem:** For any integer $latex x$, if $latex x^2$ is odd, then $latex x$ is odd.

We can assign the statements above to P (the hypothesis) and Q (the conclusion) as follows.

P: $latex x^2$ is odd

Q: $latex x$ is odd.

If we are going to change this statement to “if NOT Q then NOT P,” then, we have to find the opposite of P and opposite of Q. Continue reading