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Introduction to Abstract Mathematics

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Eligibility: CTY-level or Advanced CTY-level math score required

Prerequisites: Successful completion of Linear Algebra and Multivariable Calculus or equivalent

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

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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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 virtual classroom for discussions with the instructor. The classroom works on standard computers with the Adobe Connect Add-in or Adobe Flash plugin, and also tablets or handhelds that support the Adobe Connect Mobile appStudents who are unable to attend live sessions will need a computer with the Adobe Connect Add-in or Adobe Flash plugin installed to watch recorded meetings. The Adobe Connect Add-in, Adobe Flash plugin, and Adobe Connect Mobile app are available for free download. Students who do not have the Flash plug-in installed or enabled on their browsers will be prompted to download and install the Adobe Connect add-in when accessing the virtual classroom.