Proof by Contradiction

Proof by Contradiction is very similar to a direct proof. But, instead we try to prove a proposition false to (hopefully) derive some false statement like 1=0 showing the proposition could never be false and hence it is true.

Example:

Prove that 6n+20m ≠ 49 for any two integers n,m.

Proving this can be a bit tricky. Testing every two integers is impossible. It could be done by giving a statement about the left side being even and the right being odd; but that proof could be a bit elusive. But, showing that 6n+20m = 49 is impossible is much more straight forward.

Proof:

Suppose, by way of contradiction, the m and n are integers that satisfy 6n+20m=49. We can factor a 2 out of the left hand side to yield 6n+20m=2*(3n+10m)=49. We then divide by 2 to have 3n+10m=24.5. However, 3n+10m is an integer and 24.5 is not therefore we have derived a false statement. This shows that 6n+20m=49 can never be true and thus 6n+20m ≠ 49 for all integers.