Within mathematics proof by contrapositive essentially means that
we are saying the proposition to be false. An example of this would
be saying that p -> q then its contrapositive of ~q -> ~p to be
equivalent. We say this because p (x ) -> q ( x ) is infact true for x
then ~q ( x ) -> ~p ( x ) would also be true. This would be true
because it is way easier when proving contrapositive statement to
solve ~q (x ) -> ~p ( x ) than that of p ( x ) -> q ( x ).
With that being said proof by contradiction would be applied when a
negation of the therom statement ~p would be way easier to prove
than that of p using proof. An example of this would be that of
saying there is no largest even integer. Now using k+2. We can say
that k + 2 = (2n) + 2 = 2(n+1.) Then say that k+2 is even, however
k+2 is indeed larger than that of just k. This would be a contradiction
because the statement clearly says that k is the largest even integer.
This would lead us back to our original claim proving it to be true.
