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.
"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
The need for proof
The language of mathematics
Deductive reasoning and truth
Logic and Reasoning
Propositions, connectives, and truth tables
Logical equivalence and logical implication
Predicates and quantification
Sets and Functions
Sets and membership
Operations on sets
The Cartesian product
Functions and composite functions
Properties of functions
The Structure of Mathematical Proofs
Some proofs dissected
An informal framework for proofs
A more formal framework
Direct proof route maps
Examples from sets and functions
Examples from algebra
Examples from analysis
Direct Proof: Variations
Proof using the contrapositive
Proof of biconditional statements
Proof of conjunctions
Proof by contradiction
Existence and Uniqueness
Constructive existence proofs
Non-constructive existence proofs
Proof by induction
Variations on proof by induction
Hints and Solutions to Selected Exercises