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
