A common method to establish 
is through proof by
contradiction. As in direct proofs, we assume p is true but we also
assume the negation of q is true. From these facts, we deduce that
the negation of p is also true (or that p is false and hence the
contradiction). Our conclusion is that the original statement q must
be true. Logically we have
is equivalent to
.
A similar technique is known as reductio ad absurdum. Again, we wish to
establish . Suppose that we know r is a true statement.
As with proof by contradiction we assume p and the negation of q to
be true. Suppose we can now show that these imply that the negation of
r is true (or that r is false). We now have that r is both true
and false which is absurd. Our conclusion is that q is true.