An Elementary Transition to Abstract Mathematics will help students move from introductory courses to those where rigor and proof play a much greater role.

The text is organized into five basic parts: the first looks back on selected topics from pre-calculus and calculus, treating them more rigorously, and it covers various proof techniques; the second part covers induction, sets, functions, cardinality, complex numbers, permutations, and matrices; the third part introduces basic number theory including applications to cryptography; the fourth part introduces key objects from abstract algebra; and the final part focuses on polynomials.


  • The material is presented in many short chapters, so that one concept at a time can be absorbed by the student.
  • Two "looking back" chapters at the outset (pre-calculus and calculus) are designed to start the student’s transition by working with familiar concepts.
  • Many examples of every concept are given to make the material as concrete as possible and to emphasize the importance of searching for patterns.
  • A conversational writing style is employed throughout in an effort to encourage active learning on the part of the student.

chapter Chapter 1|5 pages

A Look Back: Precalculus Math

chapter Chapter 2|6 pages

A Look Back: Calculus

chapter Chapter 3|8 pages

About Proofs and Proof Strategies

chapter Chapter 4|8 pages

Mathematical Induction

chapter Chapter 5|5 pages

The Well-Ordering Principle

chapter Chapter 6|5 pages


chapter Chapter 7|4 pages

Equivalence Relations

chapter Chapter 8|6 pages


chapter Chapter 9|5 pages

Cardinality of Sets

chapter Chapter 10|7 pages


chapter Chapter 11|5 pages

Complex Numbers

chapter Chapter 12|8 pages

Matrices and Sets with Algebraic Structure

chapter Chapter 13|8 pages

Divisibility in Z and Number Theory

chapter Chapter 14|10 pages

Primes and Unique Factorization

chapter Chapter 15|8 pages

Congruences and the Finite Sets Z n

chapter Chapter 16|9 pages

Solving Congruences

chapter Chapter 17|5 pages

Fermat's Theorem

chapter Chapter 18|7 pages

Diffie-Hellman Key Exchange

chapter Chapter 19|6 pages

Euler's Formula and Euler's Theorem

chapter Chapter 20|7 pages

RSA Cryptographic System

chapter Chapter 21|9 pages

Groups - Definition and Examples

chapter Chapter 22|6 pages

Groups - Basic Properties

chapter Chapter 23|8 pages

Groups - Subgroups

chapter Chapter 24|6 pages

Groups - Cosets

chapter Chapter 25|7 pages

Groups - Lagrange's Theorem

chapter Chapter 26|8 pages


chapter Chapter 27|6 pages

Subrings and Ideals

chapter Chapter 28|4 pages

Integral Domains

chapter Chapter 29|6 pages


chapter Chapter 30|5 pages

Vector Spaces

chapter Chapter 31|5 pages

Vector Space Properties

chapter Chapter 32|4 pages

Subspaces of Vector Spaces

chapter Chapter 33|8 pages


chapter Chapter 34|7 pages

Polynomials - Unique Factorization