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Techniques of Mathematical Proof in Math 200, Spring 2009 - Prof. Lawrence Stout, Assignments of Mathematics

The course 'techniques of proof' for math 200 at illinois wesleyan university in the spring of 2009. The professor, lawrence stout, provides a detailed description of the course, its objectives, required texts, and grading policies. The course focuses on teaching students how to write mathematical proofs, using techniques from classical logic and various mathematical fields such as set theory, number theory, and geometry.

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Uploaded on 08/16/2009

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Math 200, Spring 2009
Techniques of Proof
2 MWF, CNS E108
Professor: Lawrence Stout
Office: C209C CNS
Office Hours: M 9-12 W 9-11
Phone: 556-3038
e-mail: lstoutiwu.edu
Web: http://www.iwu.edu/lstout
Required Texts: Leslie Lamport
The L
A
T
E
X User’s Guide and Reference Manual
2nd Edition
Course Description
This is a course on how to write mathematical proofs. Since most 300
level math courses and all 400 level ones use a lot of proof and expect you
to be able to find and write proofs it serves as one of the keystones of our
curriculum. What you will do in this course is search for proofs (both alone
in your homework and with your peers in class), critique proposed proofs to
see if they are correct, write the proofs you have found, rewrite what you
have written so that it is clear and expressed in (what passes in mathematics
for) flowing English. In addition you will learn how to typeset your work in
L
A
T
EX, the mark-up language used for producing mathematical text. All of
your grade in this course will be earned by finding and writing proofs.
This course is writing intensive, not just because you need a writing
intensive course in the major to graduate, but also because the process of
finding a proof and the process of writing it down coherently are best learned
by doing, not watching. We come to know mathematics through a many
staged process: first we explore examples; then make conjectures, look for
counterexamples, and if none are found search for a proof; eventually we
see the idea of a proof; then we fill in all of the details to be certain of the
result, often finding flaws which we need to correct; then we write the proof
so that the idea of the proof is clear, there are enough details given so that
the certainty of the result is not in doubt, and the argument flows. You will
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Math 200, Spring 2009

Techniques of Proof

2 MWF, CNS E

Professor: Lawrence Stout Office: C209C CNS Office Hours: M 9-12 W 9- Phone: 556- e-mail: lstoutiwu.edu Web: http://www.iwu.edu/∼lstout Required Texts: Leslie Lamport The LATEX User’s Guide and Reference Manual 2 nd^ Edition

Course Description

This is a course on how to write mathematical proofs. Since most 300 level math courses and all 400 level ones use a lot of proof and expect you to be able to find and write proofs it serves as one of the keystones of our curriculum. What you will do in this course is search for proofs (both alone in your homework and with your peers in class), critique proposed proofs to see if they are correct, write the proofs you have found, rewrite what you have written so that it is clear and expressed in (what passes in mathematics for) flowing English. In addition you will learn how to typeset your work in LATEX, the mark-up language used for producing mathematical text. All of your grade in this course will be earned by finding and writing proofs. This course is writing intensive, not just because you need a writing intensive course in the major to graduate, but also because the process of finding a proof and the process of writing it down coherently are best learned by doing, not watching. We come to know mathematics through a many staged process: first we explore examples; then make conjectures, look for counterexamples, and if none are found search for a proof; eventually we see the idea of a proof; then we fill in all of the details to be certain of the result, often finding flaws which we need to correct; then we write the proof so that the idea of the proof is clear, there are enough details given so that the certainty of the result is not in doubt, and the argument flows. You will

often only find the flaws in your own reasoning when you attempt to write the final version of a proof or present the proof to a peer. The argument in a proof in mathematics follows rules which have been codified in (usually) classical logic as forms of argument which will not introduce error. While the search for a proof may follow experimental, computational, or diagrammatic approaches and may make leaps of faith which leave holes to be filled in later, the exposition of a proof takes a deductive approach, starting from definitions and previously proved results and arguing to the result being proved. We will start by looking at the major techniques of proof and how they arise from standard ideas in logic. We will look at standard strategies in proof: working forward; working backwards; seeing what general features you can see in a good example. We will look at the grammar of mathematical symbolism and how mathematics gets incorporated into text. The mathematical content of the course comes from naive set theory, number theory, category theory, geometry, and calculus. Most of modern mathematics takes set theory as a starting place and makes its definitions in set theoretic terms. In our curriculum this is most apparent in probability, topology, and modern algebra. The point of the course, though, is for you to develop skill in finding and writing proofs rather than coverage of any particular mathematical content.

Written work and grading:

In this class you will find proofs, give me rough drafts, and then polish the results. You will accumulate points for this activity. There will also be an expository paper with 100 points possible. If you get 500 points I’ll guarantee an A. The minimum for a C is a consistent struggle leading to marked improvement but perhaps only 250 points. I will provide several assignment packets giving the basic definitions in an area and a collection of propositions with the instructions to give proof, or to give a counter example and a salvage of the proposition with a proof. After I hand out the packet you’ll have about two weeks to get rough drafts (each proof on a separate page) in. If at first you don’t succeed, keep trying. A reasonably common grade on a rough draft is “not yet”. Final drafts typeset in LATEXare due one week after I return the approved rough draft. Each theorem, proposition or problem has a point value on the sheet (averaging about 3 points); a correct proof in rough draft earns full credit. If what you