Proof by contrapositive proves a conditional theorem of k to l by showing that the contrapositive not l to not k is true. In this case, it is assumed that not l is true and not k is proven as a result of not l. Proof by contradiction assumes that if a conclusion that cannot be true because a theorem is false, then the theorem must be true. Here is an example of proof by contradiction: Suppose square root of 2 is rational. Integers a and b exist so that square root of 2 = a/b. Assuming that a/b is in simplest form, we multiply both sides of the equation by b and square it giving us: 2b2 = a2 Since it shows that a2 is even, then a is even. We can prove this by a = 2m (m E Z). So: 2b2 = a2 = (2m)2 = 4m2 Dividing that by 2 gives us b2 = 2m2 and shows that b2 is even. We can prove this by b = 2n (n E Z). So, if square root of 2 = a/b, then a and b must be even as well as both multiples of 2. Therefore, square root of 2 must not be rational, so square root of 2 is irrational.