MATH 215
Introduction to Advanced Mathematics (Call 24907)

Spring 2009

MWF 1:00 315 BSB 

Instructor: Henri Gillet
Office: 405 SEO
Office Hours: M,W 2-3, W 10-10:50  (Tentative: These will probably change)
phone: (312) 413-2157
e-mail: gillet@uic.edu

Text

• P. Eccles, An Introduction to Mathematical Reasoning, Cambridge University Press.

Prerequisites

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

Description

This is the first course in theoretical mathematics, and is a prerequisite to all advanced pure mathematics courses in the department. The course's primary Goal is to learn how to create and write mathematical proofs. You will learn basic techniques, such as proof by induction and contradiction. We will also learn some of the basic concepts of set theory used in advanced courses including: sets, functions, equivalence relations, cardinality and infinite sets. While the goal is not to learn topics from specific areas of mathematics, the concepts introduced will be illustrated by examples from some of the classical topics of pure mathematics such as number theory (arithmetic)

Homework

The only way to learn mathematics is by doing it, i.e., by doing problems.  Each chapter of the text has a number of exercises, the solutions to which may be found in the back of the book.  You should work through these, especially the exercises suggested below.  You will have to hand in weekly problem sets that will be graded and returned to you.  Late homework will be accepted only in exceptional circumstances (though your lowest two HW scores will be dropped).

You may discuss homework problems with other students, and you are welcome to come and talk to me about them, but you must write up your solution independently.

Most problem sets will consist of writing proofs, and you will be graded on the clarity and conciseness of your writing.

Grading

There will be 2 midterm exams and a final exam. Each midterm will count for 20% of your final grade, as will the problem sets (remember the two lowest problem sets will be dropped; the final will count for 40%.

Midterm 1: Wednesday February 18, 
Midterm 2: Wednesday  April 1, 2009

Week	Topic (Read these sections)	Additional Reading	Problems for practice	HW to be handed in (Due on Monday of the following week)	Comments
1/12 to 1/16	Ch 1 & Ch 2		Chapter 1: 1.2, 1.4, 1.5 Chapter 2: 2.1, 2.3, 2.5	homework.pdf Solutions	Propositions, predicates, truth tables, ‘and’, ‘or’, ‘not’, implication
1/21 & 1/23	Ch 3	Hints and Suggestions for writing proofs Handout on ordered fields	Chapter 3: 3.1ii), 3.5, 3.6	HW2.pdf	Techniques of proof.  Axioms for ordered fields.
1/26 to 1/30	Ch 4 and 5		Chapter 4: 4.1, 4.2, 4.5 Chapter 5: 5.2, 5.6, 5.7	Pages 54, 55: Qns: 11, 14, 17 (Extra credit: in qn 17, prove that the an form an increasing sequence with limit 5.)	Proof by Contradiction, Principle of Mathematical Induction
2/2 to 2/6	Ch 6	The empty set The number zero The symbol for the empty set. Russell’s Paradox Children’s concepts of numbers.  (This is just a randomly chosen paper – I don’t claim that it is authoritative.)	Chapter 6	Page 115, 116: Qns: 4, 8, 15	Sets. Also the quantifiers $ and "  will be introduced. And there will be some discussion of functions.
2/9 to 2/13
2/ 18 MIDTERM		Homework Solutions	Sample Mid-term
2/20 to 2/25	Ch 7	The Midterm Redo any questions you got wrong on the Mid Term.  You will get ½ credit for the difference between your score and full score on each question.  Hand in your corrections, together with your exam book on Wednesday.		Page 115, 116: Qns: 11, 12,   (Due 2/25)	Quantifiers
2/27 to 3/11	Ch 21	Modular Arithmetic	20.1, 21.3, 21.5	Page 271, Qns 1,5,6
3/11 to 3/18		Groups  See the Wikipedia article for an introduction.		Homework for next wednesday
4/1  MIDTERM		Solutions to the HW for Ch 7 and 21	Sample Midterm
Due Monday 4/14		You can cut up a ball of radius 1 and reassemble it into a ball of radius 2 Every surjective map has a right inverse (Axiom of choice)	math215_midterm2.pdf	Redo any questions you got wrong on the Mid Term.  You will get ½ credit for the difference between your score and full score on each question.  Hand in your corrections, together with your exam book on Monday.
Due Wednesday 4/15		Peano space filling curve Space filling curve applet How long is the coastline of England?	9.1, 9.4, 9.6	P 119, Qn 21, 22 P183 2, 7	Cardinality of sets
Due Wednesday 4/22				P 182, Qns 14, 17, 20
Due Wednesday 4/29		Four Color Theorem Euler’s Formula		Sample Final Questions We will go over these starting on the 29^th, and so I will not be giving extensions on handing this in for credit.
The Etymology of Pigeonhole – in 1577 it really did mean a hole for pigeons! This is a link to the OED entry for the word. It will be good for three days. After that you will need to go directly to the online OED in the UIC library reference shelf.
PDFs of handwritten solutions to the HW problems from the last 2 weeks that we did not go over in class.
Monday May 4		Review / Office hours SEO 612 10-11:30 AM		Final Exam 1:00 315 BSB