n2 - CS 70 Spring 2009 Proofs Discrete Mathematics and...

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CS 70 Discrete Mathematics and Probability Theory Spring 2009 Alistair Sinclair, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar. We all like to assert things, and few of us like to say things that turn out to be false. A proof provides a means for guaranteeing such claims. Proofs in mathematics and computer science require a precisely stated proposition to be proved. But what exactly is a proof? How do you show that a proposition is true? Recall that there are certain propositions called axioms or postulates, that we accept without proof (we have to start somewhere). A formal proof is a sequence of statements, ending with the proposition being proved, with the property that each statement is either an axiom or its truth follows easily from the fact that the previous statements are true. For example, in high school geometry you may have written two-column proofs where one column lists the statements and the other column lists the justifications for each statement. The justifications invoke certain very simple rules of inference which we trust (such as if P is true and Q is true, then P Q is true). Every proof has these elements, though it does not have to be written in a tabular format. And most importantly, the fact that each step follows from the previous step is so straightforward, it can be checked by a computer program. A formal proof for all but the simplest propositions is too cumbersome to be useful. In practice, mathemati- cians routinely skip steps to give proofs of reasonable length. How do they decide which steps to include in the proof? The answer is sufficiently many steps to convince themselves and the reader that the details can easily be filled in if desired. This of course depends upon the knowledge and skill of the audience. So in practice proofs are socially negotiated. During the first few weeks of the semester, the proofs we will write will be quite formal. Once you get more comfortable with the notion of a proof, we will relax a bit. We will start emphasizing the main ideas in our proofs and sketching some of the routine steps. This will help increase clarity and understanding and reduce clutter. A proof, for the purposes of this class, is essentially a convincing argument. Convincing to whom? First, to you, the author, second, to your classmates, third, to your professor and your TA. In this lecture you will see some examples of proofs. The proofs chosen are particularly interesting and elegant, and some are of great historical importance. But the purpose of this lecture is not to teach you about these particular proofs (and certainly not for you to attempt to memorize any of them!). Instead, you should see these as good illustrations of various basic proof techniques. You will notice that sometimes when it is hard to even get started proving a certain proposition using one proof technique, it is easy using a different technique. This will come in handy later in the course when you work on homework problems or try to prove a statement on your own. If you find yourself completely stuck, rather than getting discouraged you
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