MATH 215 - Introduction to Advanced Mathematics
University of Illinois at Chicago
Fall 2026

This course offers an introduction to methods of proofs used in different fields in mathematics.

Basic Information

Syllabus: pdf
Time and Location: M-W-F 10:00-10:50 am, 310 Addams Hall (AH)
Instructor: Lev Reyzin, SEO 417
Textbook: Peter J. Eccles, An Introduction to Mathematical Reasoning: numbers, sets and functions
Office Hours: T 11:00-11:50am (online), F 11:00-11:50am (in-person)

Exams

midterm exam: Friday, October 16, 10:00-10:50 am, during class
final exam: Friday, December 11, 10:30-12:30 am, in usual classroom

Problem Sets

problem set 1 due 9/7/26
problem set 2 due 9/21/26
problem set 3 due 10/2/26
problem set 4 due 10/12/26
problem set 5 due 10/28/26
problem set 6 due 11/09/26

Lectures and Readings

lecture 1 (8/24/26)
covered material: intro to the course, what is mathematics
optional reading: A Mathematician's Lament by Paul Lockhart

lecture 2 (8/26/26)
covered material: the language of mathematics
reading: chapter 1

lecture 3 (8/28/26)
covered material: implications
reading: chapter 2

lecture 4 (8/31/26)
covered material: direct proofs
reading: chapter 3

lecture 5 (9/02/26)
covered material: proofs by contradiction
reading: chapter 4

lecture 6 (9/04/26)
covered material: intro to proofs by induction
reading: chapter 5.1 and 5.2

lecture 7 (9/09/26)
covered material: definition by induction
reading: chapter 5.3

lecture 8 (9/11/26)
covered material: strong induction
reading: chapter 5.4

lecture 9 (9/14/26)
covered material: sets and their defitions
reading: chapter 6.1

lecture 10 (9/16/26)
covered material: set operations and the power set
reading: chapter 6.2 and 6.3

lecture 11 (9/18/26)
covered material: quantifiers
reading: chapter 7

lecture 12 (9/21/26)
covered material: functions and formulae
reading: chapter 8.1

lecture 13 (9/23/26)
covered material: function compositions, sequences, images, and graphs
reading: chapter 8.2 - 8.5

lecture 14 (9/25/26)
covered material: properties of functions
reading: chapter 9.1

lecture 15 (9/28/26)
covered material: inverses, functions on sets, Peano's axioms
reading: chapter 9.2 - 9.4

lecture 16 (9/30/26)
covered material: review of covered material so far
reading: review parts I and II

lecture 17 (10/2/26)
covered material: counting, the includsion-exclusion principle
reading: chapter 10

lecture 18 (10/5/26)
covered material: properties of finite sets
reading: chapter 11

lecture 19 (10/7/26)
covered material: counting sets of functions and sets of subsets
reading: chapter 12.1 - 12.2

lecture 20 (10/9/26)
covered material: the Pascal's triangle and the binomial theorem
reading: chapter 12.3

lecture 21 (10/12/26)
covered material: number systems
reading: chapter 13

lecture 22 (10/14/26)
covered material: midterm review
reading: review chapters 1 - 9

lecture 23 (10/16/26)
midterm exam: no lecture

lecture 24 (10/19/26)
covered material: counting infinite sets, Cantor's theorem
reading: chapter 14

lecture 25 (10/21/26)
covered material: the division theorem
reading: chapter 15

lecture 26 (10/23/26)
covered material: the Euclidean algorithm for finding greatest common divisors
reading: chapter 16

lecture 27 (10/26/26)
covered material: consequences of the Euclidean algorithm
reading: chapter 17.1 and 17.2

lecture 28 (10/28/26)
covered material: coprime pairs
reading: chapter 17.3

lecture 29 (10/30/26)
covered material: linear diophantine equations
reading: chapter 18

lecture 30 (10/2/26)
covered material: modular congruence
reading: chapter 19.1

lecture 31 (10/4/26)
covered material: the remainder map, division in congruences
reading: chapter 19.2, 19.3

lecture 32 (10/6/26)
covered material: solutions to linear congruences
reading: chapter 20.1