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 instructs students on the primary basic logic of mathematical proofs, showing how proofs of mathematical statements work. The text provides basic core techniques of how to read and write proofs through examples. The basic mechanics of proofs are provided for a methodical approach in gaining an understanding of the fundamentals to help students reach different results. A variety of fundamental proofs demonstrate the basic steps in the construction of a proof and numerous examples illustrate the method and detail necessary to prove various kinds of theorems.

Paperback, 296 Pages

Published: January 2012

Imprint: Academic Press

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…for those required to teach ‘transition courses’, I recommend perusal of this book as a possible course text."--MAA online, December 3, 2013


  • Chapter 1: Getting Started
    Introduction and Basic Terminology
    General Suggestions

    Chapter 2: Basic Techniques to Prove If/Then Statements
    What Does "If/Then" Mean?
    Direct Proof
    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
    Equality of Numbers

    Chapter 4: Some mathematical topics on which to practice proof techniques
    Basic Set Theory
    About Functions
    The Basics of Groups
    Cardinality and Sizes of Infinity

    Chapter 5: Review Exercises
    Exercises Without Solutions
    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
    Solutions for the Review Exercises

    Other Books on the Subject of Proofs and Mathematical Writing


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