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