CONTRAPOSITIVE IS A QUITE POWERFUL method in proof: it allows
you to attack a proof backwards. Instead of going from the
assumptions and trying to derive the result, you start by assuming the
result is false and show that this violates one of the assumptions.
This makes use of the logical law of *contrapositive:*

IF A THEN B |
is equivalent to | IF NOT B THEN NOT A |

Since we are assuming that the result is not true and end up by
contradicting our assumptions, this is a kind of proof by
contradiction known as **proof by contrapositive**.^{2}We used it back in Section 1 for the very first proof; it is
used quite a lot. It gives a lot of flexibility in IF AND ONLY IF, because it
allows us to interpret the ``forward'' and ``backward'' part of the
proof in several different ways:

By applying the law of contrapositive, it is fairly easy to see that all three forms are in fact equivalent. This flexibility in proof methods is yet another reason to prefer the two-step approach to if and only if proofs to the monolithic approach.