The Nuts and Bolts of Proofs

An Introduction to Mathematical Proofs


  • Antonella Cupillari, Pennsylvania State Erie, Behrend College, U.S.A.

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


Book information

  • Published: January 2012
  • ISBN: 978-0-12-382217-8


"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‚Ķfor those required to teach ‚Äėtransition courses‚Äô, I recommend perusal of this book as a possible course text."--MAA online, December 3, 2013

Table of Contents

List of Symbols

Chapter 1 Getting Started

    Introduction and Basic Terminology

    General Suggestions

Chapter 2 Basic Techniques to Prove If/Then Statements

¬†¬†¬†¬†What Does ‚ÄúIf/Then‚ÄĚ Mean?

    The Negation of a Statement: AND/OR

    Proof by Contrapositive

    Proof by Contradiction

Chapter 3 Special Kinds of Theorems

¬†¬†¬†¬†‚ÄúIf and Only If‚ÄĚ or ‚ÄúEquivalence Theorems‚ÄĚ

    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 ‚ÄúProofs‚ÄĚ

    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