Mathematical Writing - Vivaldi Franco 2014
Wrong Implications
Forms of Argument
Inappropriate handling of implications results in common mistakes. Instead of proving an implication, we may end up proving its converse; or we may assume the statement we are meant to prove, deduce from it a true statement, and believe we’ve completed the proof. These faulty deductions—of which we now show an example—are sometimes called circular arguments, or non sequiturs.6
Prove that
WRONG PROOF.
We were supposed to prove , where . Instead we have assumed , and correctly deduced from it the true statement . However, the deduction TRUE (unlike the deduction ) gives us no information about , from Table (4.9). Indeed, had we started from the false statement , we would have reached exactly the same conclusion.
There are two methods for fixing this problem.
First method: retracing the steps. We regard the chain of deductions displayed above as ’rough work’; then we start from the end and prove the chain of converse implications.
PROOF.
where in the first and the last implications we have taken the positive square root of each side. We have proved the implication , from which we deduce that is TRUE.
Clearly, we could not have come up with such a proof without having done the ’rough work’ first; the next method does not require this.
Second method: contradiction.
PROOF. Let us assume :
We have proved that , which implies that is , that is, is TRUE.