HO24 - Maggie Johnson CS103A Handout#24 Number Theory...

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Maggie Johnson Handout #24 CS103A Key topics: * Why is Proof Important? * What Are We Trying to Prove? * The Art of Proving Things * A Review of Proof Strategies * Direct Proof * Basics of Number Theory * Indirect Proof: Contradiction * More Number Theory * Quantifiers I: Construction * Quantifiers II: Counterexample and Choose * Some Famous Conjectures * More Practice Proofs * Catalog of Proof Forms and When to Use Them There are three kinds of people in the world; those who can count and those who can't. We have just spent weeks learning about first-order logic and how to do both formal and informal proofs in FOL. For the most part, the proofs we have done pertained to formal logic, e.g., we proved things about p, q, r, and Tet(a). We occasionally did a mathematical proof where we were interested in the actual meaning of the premises and the conclusion. For example, we proved the sum of two even integers is another even integer. Now, we turn our attention to mathematical proofs. Our goal is to use the foundations that we have laid in formal logic to help us derive solid proofs of meaningful statements. We will begin by doing “semi-formal” proofs in this context where we provide detailed statement and reason charts. We call such proofs semi-formal because they will not be as rigorous (and capable of mechanical checking) as the ones we did in Fitch. But our semi-formal proofs will leave out no steps. As we proceed, we will begin to work on the skills required to do informal proofs. It takes practice learning what is appropriate to leave out in an informal mathematical proof. Why is Proof Important? Most students are introduced to the concept of mathematical proof in high-school geometry. There, students learn that you have to do a proof about geometric properties because:
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1) Observation cannot prove because our eyes can deceive us: 2) Measurement cannot prove because the certainty of the conclusion we arrive at is dependent on the precision of the measuring instrument and care of the measurer. 3) Experiment cannot prove because the conclusions are only probable ones: It is probable that the dice are loaded if 10 successive 7's are thrown; it is even more probable if 20 successive 7's are thrown ( but it's not certain). This last example is especially relevant in computer science. Thus far, you have validated the correctness of the programs that you write by testing, i.e., by experimentation. In most cases however, it is impossible to test all relevant inputs. So, you do enough testing to convince yourself that the boundary cases and several cases in between are covered. This is good enough for school assignments, but if you are designing the software for collision avoidance in a new aircraft, you will want to do a lot more than just test isolated inputs; you will want to use proof techniques to verify correctness, as well as validate it. In addition, proof is important to science and engineering students because it teaches us to think in
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HO24 - Maggie Johnson CS103A Handout#24 Number Theory...

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