1st Edition

Understanding Mathematical Proof

By John Taylor, Rowan Garnier Copyright 2014
    414 Pages 62 B/W Illustrations
    by Chapman & Hall

    414 Pages
    by Chapman & Hall

    The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs.

    Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techniques that mathematicians adopt to prove their results, and offers advice and strategies for constructing proofs. It will improve students’ ability to understand proofs and construct correct proofs of their own.

    The first chapter of the text introduces the kind of reasoning that mathematicians use when writing their proofs and gives some example proofs to set the scene. The book then describes basic logic to enable an understanding of the structure of both individual mathematical statements and whole mathematical proofs. It also explains the notions of sets and functions and dissects several proofs with a view to exposing some of the underlying features common to most mathematical proofs. The remainder of the book delves further into different types of proof, including direct proof, proof using contrapositive, proof by contradiction, and mathematical induction. The authors also discuss existence and uniqueness proofs and the role of counter examples.

    Introduction
    The need for proof
    The language of mathematics
    Reasoning
    Deductive reasoning and truth
    Example proofs

    Logic and Reasoning
    Introduction
    Propositions, connectives, and truth tables
    Logical equivalence and logical implication
    Predicates and quantification
    Logical reasoning

    Sets and Functions
    Introduction
    Sets and membership
    Operations on sets
    The Cartesian product
    Functions and composite functions
    Properties of functions

    The Structure of Mathematical Proofs
    Introduction
    Some proofs dissected
    An informal framework for proofs
    Direct proof
    A more formal framework

    Finding Proofs
    Direct proof route maps
    Examples from sets and functions
    Examples from algebra
    Examples from analysis

    Direct Proof: Variations
    Introduction
    Proof using the contrapositive
    Proof of biconditional statements
    Proof of conjunctions
    Proof by contradiction
    Further examples

    Existence and Uniqueness
    Introduction
    Constructive existence proofs
    Non-constructive existence proofs
    Counter-examples
    Uniqueness proofs

    Mathematical Induction
    Introduction
    Proof by induction
    Variations on proof by induction

    Hints and Solutions to Selected Exercises

    Bibliography

    Index

    Biography

    John Taylor, Rowan Garnier

    "The book is written in a precise and clear style, with lots of appropriately chosen examples and a sufficient amount of (clear) diagrams. … could be useful to, and enjoyed by, students seeking a concise introduction to the notion of mathematical proof."
    London Mathematical Society Newsletter, No. 454, January 2016

    "The manner in which the authors expose their ideas is a very kind and easy to understand one. The book contains lots of examples and comments. Far more, all the judgements are well exposed. The examples that are offered cover a large area of elementary mathematics, such as calculus, logic, sets and functions, linear algebra, and group theory. We highly recommend this book, first of all to those who study mathematics, but we also find it useful for those who study engineering and computer science."
    Zentralblatt MATH 1311