The Nuts and Bolts of Proofs, 4th Edition

An Introduction to Mathematical Proofs

The Nuts and Bolts of Proofs, 4th Edition,Antonella Cupillari,ISBN9780123822178


Academic Press




235 X 191

This valuable resource demonstrates the construction of proofs to help students develop the skills they need to transition to higher-level, more abstract math courses.

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Key Features

  • Jumps right in with the needed vocabulary-gets students thinking like mathematicians from the beginning
  • Offers a large variety of examples and problems with solutions for students to work through on their own
  • Includes a collection of exercises without solutions to help instructors prepare assignments
  • Contains an extensive list of basic mathematical definitions and concepts needed in abstract mathematics


The Nuts and Bolts of Proofs instructs students on the primary basic logic of mathematical proofs, showing how proofs of mathematical statements work. The text provides basic core techniques of how to read and write proofs through examples. The basic mechanics of proofs are provided for a methodical approach in gaining an understanding of the fundamentals to help students reach different results. A variety of fundamental proofs demonstrate the basic steps in the construction of a proof and numerous examples illustrate the method and detail necessary to prove various kinds of theorems.


A supplement for undergraduate courses in higher mathematics as a primer to assist engineering and physical science students on fundamental proof techniques

Antonella Cupillari

Antonella Cupillari is an associate professor of mathematics at Pennsylvania State Erie in Behrend College. She received her Laurea in Mathematics in Italy, and her M.A. and Ph.D. at the State University of New York at Albany. She has been a participant in the Mathematical Association of America/National Science Foundation Institute on the "History of Mathematics and Its Use in Teaching." Cupillari is the author of several papers in analysis, mathematics education, and the history of mathematics. She is also the author of the first edition of The Nuts and Bolts of Proofs.

Affiliations and Expertise

Pennsylvania State Erie, Behrend College, U.S.A.

The Nuts and Bolts of Proofs, 4th Edition

Chapter 1: Getting Started
Introduction and Basic Terminology
General Suggestions

Chapter 2: Basic Techniques to Prove If/Then Statements
What Does "If/Then" Mean?
Direct Proof
The Negation of a Statement, And/Or
Proof by Contrapositive
Proof by Contradiction

Chapter 3: Special kinds of Theorems
"If and only if" or Equivalence Theorems
Use of Counterexamples
Mathematical Induction
Existence Theorems
Uniqueness Theorems
Composite Statements
Equality of Numbers

Chapter 4: Some mathematical topics on which to practice proof techniques
Basic Set Theory
About Functions
The Basics of Groups
Cardinality and Sizes of Infinity

Chapter 5: Review Exercises
Exercises Without Solutions
Collection of Proofs

Solutions for the Exercises at the End of the Sections and the Review Exercises
Solutions for the Exercises at the End of the Sections
Solutions for the Review Exercises

Other Books on the Subject of Proofs and Mathematical Writing

Quotes and reviews

"It is written with great accuracy and a level of enthusiasm necessary for the Herculean task of launching mathematical handle-turners into the world of mathematical thinking…for those required to teach ‘transition courses’, I recommend perusal of this book as a possible course text."--MAA online, December 3, 2013

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