This week we began our study of mathematical proof. We covered several major topics,
- Proof by example (proving an existential statement true).
- Proof by counterexample (proving a universal statement false).
- Proof by exhaustion (proving an existential statement false or a universal statement true).
- Proof by generalizing from a generic particular (instantiating a generic, arbitrarily chosen element of a set and deducing that it must have a certain property using rules of logic, definitions, and accepted rules of mathematics).
- Direct proof of a universal conditional statement (in the simplest case, instantiating a generic, arbitrarily chosen, element of the indicated universal set; assuming the hypothesis of the conditional statement is true for the instantiated element; using rules of logic, definitions, and accepted rules of mathematics to deduce the truth of the conclusion).
- Proof by contradiction (negating the statement to be proved and deducing a contradiction in the enclosing system of mathematics; rejecting the contradiction, not the system, and therefore rejecting the negated statement, and asserting the original statement)
- Proof by contraposition (instead of directly proving a universal conditional statement is true, directly prove that its equivalent contrapositive is true).
To provide material for practice we established several mathematical definitions and theorems,
- definitions of even integers, odd integers, prime integers, composite integers, rational numbers, divisibility, standard factored form, div, mod, absolute value, floor and ceiling
- theorems of factoring integers into primes, properties of even and odd numbers, divisibility, absolute value, floor and ceiling
I presented examples of the above types of proof using the mathematical definitions and theorems that were developed, as well as rules of logic, rules of algebra, etc. I used a two-column format for the proofs: the left column listing formal mathematical statements; the right column giving reasons that the mathematical statements were validly deduced from what came before.
All of this was done in five hours of lecture. The students (at least some) followed the lectures closely since they caught my mistakes and challenged the reasoning behind some of the deductions. I prepared notes for the lectures last weekend (four hours) and reviewed and updated the notes before each class (five hours). The notes are handwritten so I don’t publish them for the students–I think it is better for students to see the information presented live, record what they think is important, and read the text for the ‘written’ view of the same material.
There was a quiz at the start of the week on the material from week 2. The scores were again in a bimodal distribution, with most of the students doing very well. The homework scores were high for those who did the homework. I spent approximately six hours correcting quizzes and homework (I am teaching 61 students in two sections).
It was important to establish an agreed level of detail for the proofs. As students learn to question their knowledge of what is mathematically true and false, and why they believe it, they can begin to doubt that they understand much at all about mathematics. That’s an important awakening, but it can lead to excessive denial of conventions and excessive need for detail. I offered my examples of proofs in the lectures as templates for the level of detail that they should strive for in the class–a little more detail than they will need in the future as their understanding matures. From past experience, some will think it too much work that belabors the obvious but others will see the value of exposing more layers of the chains of reasoning that bind our mathematical foundations.