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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.

This course does not have any synchronous class meetings, but students may schedule one-on-one virtual meetings directly with the instructor to answer questions or concerns.

Virtual classrooms, and student activities in the classroom, may be recorded and added to the course as an ongoing asset for all class students to review. Videos from YouTube or other web providers may be present in the course. Video recommendations or links provided at end of videos are generated by the video host provider and are not CTY recommendations. Student contributions (e.g., projects, forum posts, etc.) may remain in the course after the student completes the course. These artifacts may be preserved to showcase student work or to continue important conversations.

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

Proctor Requirements

For enrollment starting on or after January 1, 2019, a proctor is required for this course if the student’s goal is to obtain a grade. Please review Proctor Requirements for more details.

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.

Zoom online virtual classroom
This course uses an online virtual classroom which can be used for instructor-student communication if the student has any questions about the course or curriculum. The classroom works on standard computers with the Zoom desktop client and also tablets or handhelds that support the Zoom Mobile app. Students will need a computer with the Zoom desktop client installed to watch any recorded meetings. The Zoom desktop client and Zoom Mobile App are both available for free download.