The Nuts and Bolts of Proofs book cover

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

Imprint: Academic Press

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


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 “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

        Relations

             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

        Limits

             Getting Closer

             Functions and Limits

        Sizes of Infinity

    Chapter 5 Review Exercises

        Exercises without Solutions

             General Topics

             Basic Set Theory

             About Functions

             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






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