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 firstorder 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 “semiformal” proofs in this context where we provide detailed statement and reason charts.
We call such proofs semiformal because they will not be as rigorous (and capable of mechanical
checking) as the ones we did in Fitch. But our semiformal 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 highschool geometry.
There, students learn that you have to do a proof about geometric properties because:
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View Full Document1) 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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 Fall '07
 Plummer,R
 Computer Science, Number Theory, Natural number, Prime number

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