# The Nuts and Bolts of Proofs

## An Introduction to Mathematical Proofs

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.

Audience

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

Paperback, 296 Pages

Published: January 2012

ISBN: 978-0-12-382217-8

## 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 thinkingâ¦for those required to teach âtransition coursesâ, I recommend perusal of this book as a possible course text."--MAA online, December 3, 2013

## Contents

• List of Symbols

Chapter 1 Getting Started

Â Â Â Â Introduction and Basic Terminology

Â Â Â Â General Suggestions

Chapter 2 Basic Techniques to Prove If/Then Statements

Â Â Â Â What Does âIf/Thenâ Mean?

Â Â Â Â The Negation of a Statement: AND/OR

Â Â Â Â Proof by Contrapositive

Â Â Â Â Proof by Contradiction

Chapter 3 Special Kinds of Theorems

Â Â Â Â âIf and Only Ifâ or âEquivalence Theoremsâ

Â Â Â Â Use of Counterexamples

Â Â Â Â Mathematical Induction

Â Â Â Â Existence Theorems

Â Â Â Â Uniqueness Theorems

Â Â Â Â Composite Statements

Â Â Â Â Â Â Â Â Â Multiple Hypotheses

Â Â Â Â Â Â Â Â Â Multiple Conclusions

Â Â Â Â Equality of Numbers

Chapter 4 Some Mathematical Topics on Which to Practice Proof Techniques

Â Â Â Â Basic Set Theory and Indexed Families

Â Â Â Â Â Â Â Â Â Cartesian Product of Sets

Â Â Â Â Â Â Â Â Â Indexed Families of Sets

Â Â Â Â About Functions

Â Â Â Â Â Â Â Â Â Composition of Functions

Â Â Â Â Â Â Â Â Â A Little More about Functions and Sets

Â Â Â Â Relations

Â Â Â Â Â Â Â Â Â Most Common Properties of Relations

Â Â Â Â Â Â Â Â Â More about Equivalence Relations

Â Â Â Â Â Â Â Â Â A Special Relation and More Facts about Equivalence Classes

Â Â Â Â The Basics of Groups

Â Â Â Â Â Â Â Â Â Some Properties of Binary Operations

Â Â Â Â Â Â Â Â Â Special Elements

Â Â Â Â Â Â Â Â Â When the Properties Fit Together

Â Â Â Â Â Â Â Â Â Sizes and Structures

Â Â Â Â Â Â Â Â Â Groups (mod m) and Arithmetic (mod m)

Â Â Â Â Â Â Â Â Â Permutations and Symmetric Groups

Â Â Â Â Â Â Â Â Â Isomorphism and Subgroups

Â Â Â Â Â Â Â Â Â An Important Theorem

Â Â Â Â Limits

Â Â Â Â Â Â Â Â Â Getting Closer

Â Â Â Â Â Â Â Â Â Functions and Limits

Â Â Â Â Sizes of Infinity

Chapter 5 Review Exercises

Â Â Â Â Exercises without Solutions

Â Â Â Â Â Â Â Â Â General Topics

Â Â Â Â Â Â Â Â Â Basic Set Theory

Â Â Â Â Â Â Â Â Â About Functions

Â Â Â Â Â Â Â Â Â Relations

Â Â Â Â Â Â Â Â Â The Basics of Groups

Â Â Â Â Â Â Â Â Â Limits

Â Â Â Â Â Â Â Â Â Cardinality and Sizes of Infinity

Â Â Â Â Collection of âProofsâ

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

Â Â Â Â Solutions for the Exercises at the End of the Sections

Â Â Â Â Â Â Â Â Â Chapter 2: Basic Techniques to Prove If/Then Statements

Â Â Â Â Â Â Â Â Â Chapter 3: Special Kinds of Theorems

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

Â Â Â Â Solutions for the Review Exercises

Â Â Â Â Other Books on the Subject of Proofs and Mathematical Writing

Index