© 2011

Proofs and Fundamentals

A First Course in Abstract Mathematics

  • A gentle, friendly style is used, in which motivation and informal discussion play a key role The material is presented in the way that mathematicians actually use it; good mathematical tast is preferred to overly clever pedagogy There is an important section devoted to the proper writing of proofs


Part of the Undergraduate Texts in Mathematics book series (UTM)

Table of contents

  1. Front Matter
    Pages i-xxiv
  2. Proofs

    1. Front Matter
      Pages 1-1
    2. Ethan D. Bloch
      Pages 3-45
    3. Ethan D. Bloch
      Pages 47-88
  3. Fundamentals

    1. Front Matter
      Pages 89-89
    2. Ethan D. Bloch
      Pages 91-128
    3. Ethan D. Bloch
      Pages 129-169
    4. Ethan D. Bloch
      Pages 171-194
    5. Ethan D. Bloch
      Pages 195-248
  4. Extras

    1. Front Matter
      Pages 249-249
    2. Ethan D. Bloch
      Pages 251-322
    3. Ethan D. Bloch
      Pages 323-340
  5. Back Matter
    Pages 341-359

About this book


This textbook is designed to introduce undergraduates to the writing of rigorous mathematical proofs, and to fundamental mathematical ideas such as sets, functions, relations, and cardinality. The book serves as a bridge between computational courses such as calculus and more theoretical courses such as linear algebra, abstract algebra, and real analysis.

This second edition has been significantly enhanced, while maintaining the balance of topics and careful writing of the previous edition. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material such as sets, functions and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics and sequences, and suggests avenues for independent student explorations.

A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and in writing are never compromised.

Reviews of the first edition:

This is a well-written book, based on very sound pedagogical ideas. It would be an excellent choice as a textbook for a 'transition' course.
—Zentralblatt Math

'Proofs and Fundamentals' has many strengths. One notable strength is its excellent organization... There are large exercise sets throughout the book... the exercises are well integrated with the text and vary appropriately from easy to hard... Perhaps the book’s greatest strength is the author’s zeal and skill for helping students write mathematics better.
—MAA Online


Cardinality Formal Logic Introduction to Proofs combinatorics

Authors and affiliations

  1. 1., Department of MathematicsBard CollegeAnnandale-on-HudsonUSA

About the authors

Dr. Ethan D. Bloch of Bard College is the author of two Springer publications "A First Course in Geometric Topology and Differential Geometry," and the first and second editions of, "Proofs and Fundamentals: A First Course in Abstract Mathematics." More information about Dr. Ethan D. Bloch can be found on his person web page:

Bibliographic information

Industry Sectors
Finance, Business & Banking


From the book reviews:

“The contents of the book is organized in three parts … . this is a nice book, which also this reviewer has used with profit in his teaching of beginner students. It is written in a highly pedagogical style and based upon valuable didactical ideas.” (R. Steinbauer, Monatshefte für Mathematik, Vol. 174, 2014)

“Books in this category are meant to teach mathematical topics and techniques that will become valuable in more advanced courses. This book meets these criteria. … This book is well suited as a textbook for a transitional course between calculus and more theoretical courses. I also recommend it for academic libraries.” (Edgar R. Chavez, ACM Computing Reviews, February, 2012)

“This is an improved edition of a good book that can serve in the undergraduate curriculum as a bridge between computationally oriented courses like calculus and more abstract courses like algebra.” (Teun Koetsier, Zentralblatt MATH, Vol. 1230, 2012)