An Elementary Transition to Abstract Mathematics  book cover
1st Edition

An Elementary Transition to Abstract Mathematics

ISBN 9780367336936
Published October 28, 2019 by CRC Press
292 Pages 12 B/W Illustrations

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Book Description

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.

Table of Contents

A Look Back: Precalculus Math

A Look Back: Calculus

About Proofs and Proof Strategies

Mathematical Induction

The Well-Ordering Principle


Equivalence Relations


Cardinality of Sets


Complex Numbers

Matrices and Sets with Algebraic Structure

Divisibility in Z and Number Theory

Primes and Unique Factorization

Congruences and the Finite Sets Zn

Solving Congruences

Fermat’s Theorem

Diffie-Hellman Key Exchange

Euler’s Formula and Euler’s Theorem

RSA Cryptographic System

Groups-Definition and Examples

Groups-Basic Properties



Groups-Lagrange’s Theorem


Subrings and Ideals

Integral Domains


Vector Spaces

Vector Space Properties

Subspaces of Vector Spaces


Polynomials-Unique Factorization

Polynomials over the Rational, Real and Complex Numbers

Suggested Solutions to Selected Examples and Exercises

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Gove Effinger received his Ph.D. in Mathematics from the University of Massachusetts (Amherst) in 1981 and subsequently taught at Bates College for 5 years and then Skidmore College for 29 years. He is the author of two books: Additive Number Theory of Polynomials over a Finite Field (with David R. Hayes), and Common-Sense BASIC: Structured Programming with Microsoft QuickBASIC (with Alice M. Dean), as well as numerous research papers. His research focus has primarily been concerned with the similarities of the ring of polynomials over a finite field to the ring of ordinary integers.

Gary L. Mullen is Professor of Mathematics at the Pennsylvania State University, University Park, PA. He has taught both undergraduate and graduate courses there for over 40 years. In addition, he has written more than 150 research papers and five books, including both graduate as well as undergraduate textbooks. He also served as department head for seven years and has served as an editor on numerous editorial boards, including having served as Editor-in-Chief of the journal Finite Fields and Their Applications since its founding in 1995.