Math 215: Introduction to Advanced Mathematics

David Dumas

University of Illinois at Chicago
Spring 2013

Triomino tilings of an 8x8 grid with one square removed.

Basic course information

Meeting time MWF 10:00 - 10:50am in Taft Hall 316
Instructor David Dumas (
Office hours Monday 11-12 and Wednesday 12:30-1:30 in SEO 503
TA Hexi Ye (
TA Office Hours Monday 5-6pm and Friday 1-2pm in the MSLC
Monday and Wednesday 12-1pm in SEO 401
Note: Math 215 help is available in the MSLC (SEO 430) from 8am to 6pm every weekday during the semester.
CRN 34246
Textbook P. Eccles, An Introduction to Mathematical Reasoning
Cambridge University Press, 1998.
ISBN: 0521597188
Homework Policy Weekly homework will be collected in class and graded. The two lowest grades will be dropped when computing your overall homework grade.
Late homework is not accepted.
Clarity is essential. All solutions must be written in complete grammatical sentences. Submitted work must be legible (clearly hand-written or typed) and there should be no extraneous marks, symbols, or words on the pages except for your solution. Your grade will reflect both the mathematical correctness and the clarity of your writing.
You may discuss the homework problems with other students, but you must write your solutions independently.

Course Materials

Course description

The primary goal of this course is to learn how to create and write mathematical proofs. We will discuss basic proof techniques such as induction and proof by contradiction. We will also introduce some important mathematical concepts that are used in many advanced math courses, including sets, functions, equivalence relations, and cardinality.

We will cover most of parts I–IV and some of part V in the textbook. Details can be found in the weekly schedule below.


There will be two in-class midterm exams and a cumulative final exam.
Exam 1 Friday, March 1 In class
Exam 2 Friday, April 5 In class
Final Exam Friday, May 10, 10:30am-12:30pm Location TBA


Your course grade will be determined on the following basis:
Homework 15%
Exam 1 25%
Exam 2 25%
Final Exam 35%


Weekly schedule and reading assignments

Be sure to check this schedule on a regular basis; minor changes may be made during the semester according to the pace at which we cover the material. Subsets of finite sets, counting problems for functions, injections, subsets.
Read Chapter 12.
Week 1 Statements, connectives, truth tables, negation, implication.
Read Chapters 1 and 2.
Week 2 Direct proofs, axioms for groups and for ordered fields.
Read Chapter 3.
Week 3 Proof techniques: cases, contrapositive, contradiction. Examples.
Read Chapter 4.
Week 4 Proof by induction, Fibonacci numbers.
Read Chapter 5.
Week 5 Sets, subsets, equality, power set, union, intersection, difference, complement.
Read Chapter 6.
Week 6 Cartesian product, quantifiers, alternation of quantifiers, functions.
Read Chapter 7.
Week 7 Functions, graphs, composition, sequences.
Read Chapter 8.
Exam 1 on Friday.
Week 8 Injective, surjective, bijective functions, inverse functions.
Read Chapter 9.
Week 9 Images and inverse images, partitions, equivalence relations.
Read Chapter 22.
Week 10 Counting finite sets, inclusion-exclusion and pigeonhole principles.
Read Chapters 10 and 11.
Spring break
Week 11 Counting for finite sets: subsets, functions, injections, surjections.
Read Chapter 12.
Exam 2 on Friday.
Week 12 Binomial coefficients and the binomial theorem.
More on Chapter 12.
Week 13 Infinite sets, denumerable sets, countability of the rationals, uncountability of the reals.
Read Chapter 14.
Week 14 Unions of countable sets, transcendental numbers, division theorem, Euclid's algorithm.
Read Chapters 15 and 16.
Week 15 GCD and integer linear combinations, linear diophantine equations.
Read Section 17.1 and Chapter 18.
Up: Home page of David Dumas