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.