Course Description -- Math 215 -- Fall 2009

Instructor: Louis H. Kauffman

Office: 533 SEO

Phone: (312) 996-3066

E-mail: kauffman@uic.edu

Web page: http://www.math.uic.edu/~kauffman

Text Book: An Introduction to Mathematical Reasoning, Cambridge Univ. Press, by P. Eccles.

Second Text Book: What is Mathematics?, Oxford Univ. Press, by Richard Courant and Herbert Robbins. This book is available via Amazon.

Office Hours: 2:15PM to 3:00PM on MWF.

Prereqisites: Grade of C or better in MATH 181 and approval of the department.

Description: This is a first course in theoretical mathematics. It is a prerequisite to all advanced theoretical courses in the department. The Primary Goal of the course is to learn how to create and write mathematical proofs. We will introduce basic proof techniques, like proofs by induction and contradiction. We will also learn some basic mathematics that will be used in many advanced courses including: sets, functions, equivalence relations, cardinality and infinite sets. As time permits, we will cover most of Parts I-III and parts of Part V of the text.

The course will proceed via a projects and problems. We will begin by using the text by Eccles to work on basics of logic, sets, proof by induction and problems involving sets and arithmetic. Later we will branch out with a series of problems involving games played on graphs. In playing these games we directly encounter a number of mathematical phenomena and we will all work on finding proofs that these phenomena are actually happening!

Because the course is structured through evolving problem sets and projects, it is essential that you WORK ON THE PROBLEMS AND ATTEND ALL THE CLASSES. There is no recourse here to staying away from class. If you miss a class, please see the instructor and get a recapitulation of what went on in that class. Credit in the course will be a function of your homework, projects, the hour exams and the final exam. In this course HOMEWORK IS OF PRIMARY IMPORTANCE.

Keep watching this webpage for problems and notes related to the course.

See ASSIGNMENTS. This is a list of all problems assigned in the FALL 2009 course.

See Set Theory. This is an appendix from the book "Topology and Geometry" by Glen Bredon. The appendix is a concise but complete synopsis of basic set theory and goes beyond what is in Eccles. In particular, you will find discussion of well-ordering of sets, and in Theorem B18 the equivalence of a number of set theoretic principles such as well-ordering and the axiom of choice. You will also find a proof of the Cantor-Schroeder-Bernstein Theorem (B15 and B20). You do not need to read this whole appendix, but I DO ask you to look at Theorem B27. There you will find a beautiful proof that the real numbers have the same cardinality as P(N) where N is the natural numbers. The proof includes a proof that the set of finite subsets of N is countable (can you give an independent proof of this fact?). The proof of Theorem 27 uses the concept of continued fractions. We shall discuss continued fractions in class.

See Sample Problems. This is a list of sample problems for the course as a whole. You should find it useful in studying for the exam on Friday, November 20 and for the final exam in the course.

See HyperGame. This is a Supplement on the HyperGame Paradox from the book "Satan, Cantor and Infinity" by Raymond Smullyan.

See Halt!. This is a short proof of the undecidablity of the halting problem for computer programs (originally discovered by Turing). This proof (due to LK) uses reasoning analogous to the analysis of HyperGame in the supplement above.

See Supplement. This is a Supplement on Finite and Infinite Sets to be used with Assignment Number 10.

See Conway Ordinals. This is an excerpt about infinite ordinals from a book by John Horton Conway.

See Music. This is an article on musical scales by Ian Stewart from his book "Another Fine Math You've Got Me Into ...".

See Quiz. This is the first quiz with solutions.

See Exam1. This is the first exam with solutions.

See Euler's Mathematics This is an excerpt from a recent book introducing proofs and mathematical ideas.

See Euler's Formula. This is a 2-page description of Euler's formula V - E + F = 2 for connected plane graphs. Euler's formula is interesting in its own right and it is a powerful tool for solving many combinatorial problems.

BELOW YOU WILL SEE SOME OF THE PROJECTS AND PROBLEMS FROM THE PREVIOUS COURSE.

See Sprouts. These prolems begin with the graphical game of "sprouts", problems about mathematical induction and problems related to Euler's formula for plane graphs.

See Conway's Sprouts. This is an excerpt from the book "Winning Ways for Your Mathematical Plays" by Conway, Berlekamp and Guy. It contains more information about sprouts.

See Logic and Switching Circuits. This is a discussion of problems relating logic and switching circuits.

See Problems. This is a list of all problems assigned in the Spring 2009 course.

See Sample Problems. This is a list of sample problems for the course as a whole.

See TakeHomeExam for the instructions for the essay on Logic, Sets and Mathematics that was due during the Spring 2009 course.

See Axioms. This is a list of the axioms that apply to intgers to rational numbers and to real numbers. These axioms are in the book on pages 18-19 and page 24. We put them together here for your reference.

See Peano Axioms for a list of the Peano axioms for the natural numbers and a list of problems to prove, using these axioms.

See KELLEY Excerpt on basic mathematics from book by John Kelley, part of the telvision course "Continental Classroon" that Kelley gave in 1960.

See Wang Algebra for a clever approach to graphs and spanning trees.

SRPING 2009 FINAL EXAM USE THIS FOR REVIEW!