Suppose that you wish to prove the statement “If B is both open and closed, then B = ∅ or B = X.” Use tautology (p) of Example to state two different equivalent statements that could be proved instead.

 

We have now discussed three ways of proving a statement of the form p ⇒ q:

 

(i) Assume statement p and deduce statement q.

 

(ii) Assume ~ q and deduce ~ p. (Prove the contrapositive.)

 

(iii) Assume both p and ~ q and deduce a contradiction. These are the most common forms of mathematical proofs, except for proofs by mathematical induction. But before we close this chapter on logic and proof, a few informal comments are in order. In formulating a proof it is important that a mathematician (that includes you!) be very careful to use sound logical reasoning. This is what we have tried to help you develop in this chapter. But when writing down a proof it is usually unnecessary–and often undesirable –to include all the logical steps and details along the way. The human mind can only absorb so much information at one time. It is necessary to skip lightly over the steps that are well understood from previous experience so that greater attention can be focused on the part that is really new. Of course, the question of what to include and what to skip is not easy and depends to a considerable extent on the intended audience. The proofs included here will tend to be more complete than those in more advanced books or research papers, since the reader is presumably less sophisticated. As a student, you should also practice filling in more of the details, if for no other reason than to make sure that the details really do fill in. (At least be prepared to show your instructor why your “clearly” is clear and your “it follows that” really does follow.) In Sections 1 and 2 we introduced several notational symbols: ~, ∧, ∨, ⇒, ⇔, iff, ∀, ∃, and †. These are helpful in analyzing logical arguments and we shall sometimes use them in the hints given in the back of the chapter. But they are only for shorthand, and they typically should not be used in a final formal proof. Yes, mathematicians need to be able to write complete English sentences—with subjects, verbs, and periods. You will have the opportunity to read and write a great many proofs. Make the most of it! When you read a proof, analyze its structure. See what tautologies, if any, have been used. Note the important role that definitions play. Often a proof will be little more than unraveling definitions and applying them to specific cases. From time to time we shall point out the method to be used in a proof to help you see the structure that we shall be following. Finally, when you begin to write proofs yourself, do not get discouraged when your instructor returns them covered with comments and corrections. The writing of proofs is an art, and the only way to learn is by doing.