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: An Introduction to Mathematical Proofs provides basic logic of mathematical proofs and shows how mathematical proofs work. It offers techniques for both reading and writing proofs. The second chapter of the book discusses the techniques in proving if/then statements by contrapositive and proofing by contradiction. It also includes the negation statement, and/or. It examines various theorems, such as the if and only-if, or equivalence theorems, the existence theorems, and the uniqueness theorems. In addition, use of counter examples, mathematical induction, composite statements including multiple hypothesis and multiple conclusions, and equality of numbers are covered in this chapter. The book also provides mathematical topics for practicing proof techniques. Included here are the Cartesian products, indexed families, functions, and relations. The last chapter of the book provides review exercises on various topics. Undergraduate students in engineering and physical science will find this book invaluable.


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

List of Symbols

Chapter 1 Getting Started

    Introduction and Basic Terminology

    General Suggestions

Chapter 2 Basic Techniques to Prove If/Then Statements

    What Does “If/Then” Mean?

    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

         Multiple Hypotheses

         Multiple Conclusions

    Equality of Numbers

Chapter 4 Some Mathematical Topics on Which to Practice Proof Techniques

    Basic Set Theory and Indexed Families

         Cartesian Product of Sets

         Indexed Families of Sets

    About Functions

         Composition of Functions

         A Little More about Functions and Sets


         Most Common Properties of Relations

         More about Equivalence Relations

         A Special Relation and More Facts about Equivalence Classes

    The Basics of Groups

         Some Properties of Binary Operations

         Special Elements

         When the Properties Fit Together

         Sizes and Structures

         Groups (mod m) and Arithmetic (mod m)

         Permutations and Symmetric Groups

         Isomorphism and Subgroups

         An Important Theorem


         Getting Closer

         Functions and Limits

    Sizes of Infinity

Chapter 5 Review Exercises

    Exercises without Solutions

         General Topics

         Basic Set Theory

         About Functions


         The Basics of Groups


         Cardinality and Sizes of Infinity

    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

         Chapter 2: Basic Techniques to Prove If/Then Statements

         Chapter 3: Special Kinds of Theorems

         Chapter 4: Some Mathematical Topics on Which to Practice Proof Techniques

    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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