Understanding Mathematical Proof

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