Course Description
-- Math 215 -- Fall 2011Instructor: 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.
Recommended Book: What is Mathematics?, Oxford Univ. Press, by Richard Courant and Herbert Robbins. This book is available via Amazon.
Recommended Reading: Logicomix, Bloomsbury Press, by Apostolos Doxiadis and Christos H. Papadimitriou. This book is available via Amazon. Logicomix is a graphic novel about the search for foundations of mathematics at the hands of Bertrand Russell, Gotlob Frege, David Hilbert, Kurt Goedel, Ludwig Wittgenstein and others, set in the context of a political lecture by Bertrand Russell at the beginning of World War II.
Recommended Reading: The No Sided Professor. This is a topological short story by Martin Gardner.
Recommended Reading: The Feeling of Power. This is a short story by Isaac Asimov.
Recommended Reading: InfiniteHotel. A cartoon story about an infinite hotel that can seemingly accomodate any new influx of guests.
Recommended Reading: Math SciFi. Greg Egan's Homepage
Recommended Reading: Luminous Scan of a story by Greg Egan speculating about the relationship of mathematics and physical reality.
Recommended Reading: Math SciFi. "And He Built a Crooked House", by Robert Heinlein.
Office Hours: 3:15PM to 4PM 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 and the course will cover symbolic logic, basic set theory and some selected topics in combinatorics, geometry, topology and number theory. As time permits, we will cover most of Parts I-III and parts of Part V of the text.
The course will proceed via 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.
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 and the exams. There will be a final project that is due on the Wednesday before Thanksgiving.
See ASSIGNMENTS Here are Assignments 1,2,3 and 4. You are asked to read Chapters 1,2,3,4,5,6,7 of Eccles and to work on problems related to these Chapters. The assignment sheets also contain notes about Sprouts, a graphical game that we will discuss in class. This first problem set is due on Wednesday, August 31, 2011. The second problem set is due on Friday, September 9, 2011, the third assignment is due on Monday, September 19, 2011, and the fourth assignment is due on Wednesday, September 28, 2011. New Homework for the fifth assignment will appear by September 29. The fifth assignment is due on Wednesday, October 12, 2011. The sixth assignment is due Friday,October 28,2011. The seventh assignment is due on Monday, November 7,2011.
See TAKE HOME EXAM. This take home exam is due on November 28. The exam has problems similar to the homework (in fact you can look at it as another homework assignment, but it will count as an exam.). This take-home-exam will asks you to do some exploratory work. I have chosen this rather than a final paper because the demands that it makes are similar to the demands that have been made previously in the course, and you should be able to do well!
See Complex Numbers and Matrices. Please read the first sixteen pages of these notes on complex numbers and matrices. In particular, look at the exercise about matrices on page 11. Continue this exercise as follows. Let M = [[a,b], [c,d]] denote the matrix whose rows are [a,b] and [c,d]. Define the determinant of M by the formula, det(M) = ad - bc. Show that if M and N are 2 x 2 matrices and MN denotes their matrix product, then det(MN) = det(M)det(N). We will use these matrix properties in class.
See Cantor-Bernstein-Schroeder Theorem This is a PlanetMath article. See also the Wikipedia article on the Cantor-Bernstein-Schroeder Theorem. The CSB Theorem states that if there exist injections f:A ----> B and g:B -----> A for two sets A and B, then there is a bijection between A and B. Thus if |A| <= |B| and |B|<= |A|, then |A| = |B|.
See Exam2. This is a previous second exam. Our second exam will be on Wednesday, November 9, 2011. The second exam covers Eccles Chapters 1,2,3,4,5,6,7,8,9, and 14 and it is based on Homeworks 1-7.
See Exam1. This is a previous first exam with solutions. Our first exam will be on Monday, October 3, 2011. The first exam covers Eccles Chapters 1,2,3,4,5,6 and Homeworks 1,2, 3 and 4.
See Logic and Switching Circuits. This is a discussion of problems relating logic and switching circuits. The discussion includes the solution to the switching circuit problem for Assignment Number 4 (Problem 1) and includes a second problem that is part of Assignment Number 5.
See Conway's Army for a short article explaining a solitaire game and a proof about its limitations.
See Existence for an existence proof and a story related to the question whether existence proofs should exist.
See Desargues for a short proof of the three-circles theorem using the Desargues configuration. This is an elegant sample of projective geometry.
See Peano Axioms for a list of the Peano axioms for the natural numbers and a list of problems to prove, using these axioms. We will discuss the problems in class after learning more about induction.
See Geometry Here is a short set of notes on Euclidean geometry.
See Conway's Sprouts. This is an excerpt from the book "Winning Ways for Your Mathematical Plays" by Conway, Berlekamp and Guy. It contains information about the game of sprouts.
See Boolean Notation. A short article about Boolean Algebra notation.
See Peirce. A excerpt from the writings of Charles Sanders Peirce, American philosopher and mathematician (1839-1914). In this article, Peirce gives his "sign of illation", a symbol for implication that is a combination of the Boolean negation sign and the Boolean plus sign. He also has a worthwhile discussion of the nature of logic.
See Boolean Algebra. A short article about Boolean Algebra and some ideas related to it, including switching circuits, Laws of Form, recursions and circuits that count.
Keep watching this webpage for problems and notes related to the course.
BELOW THIS POINT YOU WILL FIND LINKS AND PROBLEMS FROM PREVIOUS INCARNATIONS OF THIS COURSE!
See ASSIGNMENTS 1,2,3,4. This is a list of all problems assigned in the FALL 2010 course including Assignment Number 4.
See ASSIGNMENT 5. This is Assignment Number 5.
See ASSIGNMENT 6. This is Assignment Number 6.
See ASSIGNMENT 7. This is Assignment Number 7.
See Conway Ordinals. This is an excerpt about infinite ordinals from a book by John Horton Conway.
See Supplement. This is a Supplement on Finite and Infinite Sets.
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.
See Discrete Calculus. An article about discrete calculus by Brian Hamrick.
See Discrete Calculus. An article about discrete calculus by Lou Kauffman.
See Boolean Notation. A short article about Boolean Algebra notation.
See Boolean Algebra. A short article about Boolean Algebra and some ideas related to it, including switching circuits, Laws of Form, recursions and circuits that count.
See Exam2. Second Exam with solutions.
See SampleExam2. A practice version for the second hour exam.
See Even/Odd. An article by David Joyce about Even and Odd Numbers.
See Peano Axioms for a list of the Peano axioms for the natural numbers and a list of problems to prove, using these axioms. Compare these axioms with the discussion in the article by David Joyce on Even and Odd.
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See ASSIGNMENTS. This is a list of all problems assigned in the FALL 2009 course.
See Reflections. This is a short review of the course and a list of theorems that you should be able to prove on the final exam.
See Partitions of Integers. This is an article on partitions that is relevant to the homework problem about partitions.
See Beyond Cantor. This is a separate version of Professor Squarepunkt's lecture on the Cantor Diagonal Process and a discussion of his ideas.
In Spring 2010, L. Kauffman taught Graph Theory MCS-423 at 10AM in Adams Hall 302 on MWF. Math 215 is a sufficient prerequisite for the course. The course covers basic concepts of graph theory including Eulerian and hamiltonian cycles, trees, colorings, connectivity, shortest paths, minimum spanning trees, network flows, bipartite matching, planar graphs. See Bondy and Murty. The book "Graph Theory with Applications" by Bondy and Murty was used for the course. It is freely available on the web at the above link.
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 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.
BELOW YOU WILL SEE SOME OF THE PROJECTS AND PROBLEMS FROM ANOTHER 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!
The structure of the exam is as follows. There are five questions that you are required to do. These questions involve things that you will know well, and it is mostly a matter of calmly thinking them through and writing out the proofs or solutions. Then you are asked to do ONE more problem, chosen from several more problems stated on the exam. These problems involve some of the special things that we did in the spring of 2009 like sprouts, Euler formula, something about complex numbers, possibly switches, or combinatorial coefficients, etc. But there are a number of problems to choose from and I am sure that you will find one of them that you will feel comfortable working with. Study by thinking through those parts of the course that you like and those parts that you have found challenging. Please use this time to work on your understanding of proofs and ideas.
See FinalExam.This is a copy of our final with selected solutions to some of the problems. If you have questions about how to solve these problems contact the instructor.