How can you show a mathematical statement is false?

You can show (prove) a mathematical statement to be false, by assuming it is true and then showing some logistical flaw if it were to be true. An example would be when trying to prove that the square root of 2 is irrational (i.e. cannot be expressed as a fraction where a and b are integers). You would first assume it was rational, and then a contradiction would arise which would allow you to determine that the square root of 2 cannot in fact be rational.

A simpler way, however, for example, if I said x=3 and you knew 6x=12, would be to say if x=3 6x=18, however, 6x=12, therefore x cannot equal 3.