Proving by contradiction is actually really cool. Instead of proving by cases, or proving directly, you basically take the statement, try and work out what would happen if the negative of this statement were true, and then BAM, you’ve got a proof. It usually takes the form of “If this were true, THE WORLD COULD END.” Okay, not really, but it’s along those lines.
Say we take the statement: A number cannot both be odd and even. To prove by contradiction, we would start out and say “Okay, what would happen if a number was odd and even? What would it look like?” To that you can define what it is to be odd and what it is to be even, in which lies a contradiction!
An odd number is: m ∈ ℤ, such that x = 2m + 1
An even number is: m ∈ ℤ, such that x = 2m.
The contradiction is as follows. x cannot equal both 2m, and 2m + 1 at the same time, because, when they are set equal to each other (replacing x with 2m), 2m ≠ 2m +1.
That was almost a cheat of a contradiction proof though. They are usually never this easy, but at least you get the point now!