# The Nuts and Bolts of Proofs

An Introduction to Mathematical Proofs

By

• 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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### Audience

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

### 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

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

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

Composition of Functions

A Little More about Functions and Sets

Relations

Most Common Properties of 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

Limits

Getting Closer

Functions and Limits

Sizes of Infinity

Chapter 5 Review Exercises

Exercises without Solutions

General Topics

Basic Set Theory

Relations

The Basics of Groups

Limits

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

Index