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Introduction to Abstract Mathematics

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Prerequisites: Qualifying math score and successful completion of Linear Algebra and Multivariable Calculus

Course Format: Individually Paced

Course Length: Typically 6 months

Recommended School Credit: One full year of high school credit or one semester of college credit equal to or greater than an AP class

Course Code: IAM

Course Description


Introduction to Abstract Mathematics is an online and individually-paced college course taken after Linear Algebra and Multivariable Calculus. This course teaches a student how to construct logical arguments in the form of a proof to verify mathematical statements. Techniques and methods of proofs are taught through specific examples in set theory, group theory, and real analysis. This course introduces students to the logical and rigorous mathematical foundation which all higher level math courses require. Definitions and proofs will be stressed throughout the course. Online course materials supplement the required textbook.

Each student is assigned to a CTY instructor to help them during their course. Students can contact their instructor via email with any questions or concerns at any time. Live one-on-one online review sessions can be scheduled as well to prepare for the graded assessments, which include quizzes, homework, presentations, midterm exams, and a cumulative final. Instructors use virtual classroom software allowing video, voice, text, screen sharing and whiteboard interaction.

Topics include:

  • logical reasoning
  • set theory
  • properties of functions
  • binary operations and relations
  • properties of the Integers
  • countable and uncountable sets
  • real and complex numbers
  • unique factorization of polynomials

For a detailed list of topics, click the "List of Topics" tab.

Intro to Abstract Math classroom screenshot
A sample screenshot of the Introduction to Abstract Mathematics online classroom.

Materials Needed

A textbook and supplementary book are required for this course:

An Introduction to Abstract Mathematics by Robert J. Bond and William J. Keane. Published by Waveland Press.

  • ISBN-10: 1-57766-539-2
  • ISBN-13: 978-1-57766-539-7

Intro to Abstract Math textbook

Journey through Genius: The Great Theorems of Mathematics [Paperback] by William Dunham. Published by Penguin Books USA Inc.

  • ISBN-10: 014014739X
  • ISBN-13: 978-0140147391

Journey through Genius book cover

List of Topics

Upon successful completion of the course, students will be able to demonstrate mastery over the following topics:

Mathematical Reasoning

  • Statements
  • Compound Statements
  • Implications
  • Contrapositive and Converse


  • Sets and Subsets
  • Combining Sets
  • Collections of Sets


  • Definition and Basic Properties
  • Surjective and Injective Functions
  • Composition and Invertible Functions

Binary Operations and Relations

  • Binary Operations
  • Equivalence Relations

The Integers

  • Axioms and Basic Properties
  • Induction
  • The Division Algorithm and Greatest Common Divisors
  • Primes and Unique Factorization
  • Congruences
  • Generalizing a Theorem

Infinite Sets

  • Countable Sets
  • Uncountable Sets, Cantor's Theorem, and the Schroeder-Bernstein Theorem
  • Collections of Sets

The Real and Complex Numbers

  • Fields
  • The Real Numbers
  • The Complex Numbers


  • Polynomials
  • Unique Factorization
  • Polynomials over C, R, and Q

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Sample Video

In addition to the textbooks, the course uses videos, websites, interactives and live presentations to help students learn the materials.

Sample video about former CTY student and Fields Medal Winner Terry Tao:

Fields Medal Winner Terence Tao

Sample screen shot of a quiz:
Sample quiz screenshot


Sample interactive using virtual circuits to understand logic:

Logic Lab interactive

Technical Requirements

This course requires a properly maintained computer with high-speed internet access and an up-to-date web browser (such as Chrome or Firefox). The student must be able to communicate with the instructor via email. Visit the Technical Requirements and Support page for more details.

This course uses an online classroom for individual or group discussions with the instructor. The classroom works on standard computers with the Adobe Flash plugin, and also tablets or handhelds that support the Adobe Connect Mobile app.