How do you prove a mathematical statement via contradiction?
For a proof via contradiction, you would start by assuming the statement is actually false1 (even if the given statement seems inherently correct, it is essential for the proof). Now that you have a basis for your argument, you can make use of mathematical steps and logic2 to show that the assumption leads to an impossible situation/conclusion3 (either a contradiction of the assumption or a contradiction of a fact that is known to be true). You are now able to conclude that your assumption was incorrect, and thus the original statement is true4.For example, to prove 'If x2 is even, then x is also even' via contradiction: 1) Assume the statement is actually false. Suppose that x2 is an even integer, and that x is in fact an odd integer. 2) Use mathematical steps and logic. We can represent x, being odd, as x=2n+1 where n is any integer. Now x2=(2n+1)2=4n2+4n+1. By factorising this result we end up with x2= 2k+1, where k=2n2+2n. 3) A contradiction or an impossible situation. Having shown x2=2k+1, it is impossible for x2 to be an even integer. This contradicts our original statement. 4) Conclusion. Our assumption (If x2 is even, then x is odd) has been shown to be impossible, thus proving by contradiction that the original statement (If x2 is even, then x is also even) must be correct/true.
