Proof, Logic, and Conjecture: The Mathematician's Toolbox

Description

Starting with an explanation of what 'proof' means to a mathematician, this student-friendly introductory text is aimed at undergraduates in mathematics and related disciplines. It shows students how to read and write mathematical proofs and describes how mathematicians investigate problems and formulate conjecture. Students develop their skills in logic by following precise rules, and examples and exercises relating to discovery and conjecture appear throughout. The author also covers mathematical concepts such as real and complex numbers, relations and functions and set theory. Frontmatter Table of Contents Preface Unit 1 Logic and Proofs Chapter 1 Introduction 1.1 Knowledge and Proof Proofs in Science Proofs in Law 1.2 Proofs in Mathematics Discovery and Conjecture in Mathematics Organization of the Text Chapter 2 Propositional Logic 2.1 The Basics of Propositional Logic 2.2 Conditionals and Biconditionals Biconditionals 2.3 Propositional Consequence; Introduction to Proofs Chapter 3 Predicate Logic 3.1 The Language and Grammar of Mathematics 3.2 Quantifiers 3.3 Working with Quantifiers Negations of Statements with Quantifiers Some Abbreviations for Restricted Quantifiers 3.4 The Equality Relation; Uniqueness Uniqueness Chapter 4 Mathematical Proofs 4.1 Different Types of Proofs Formal Proofs A General-Purpose Axiom System for Mathematics Informal Proofs Good Proofs 4.2 The Use of Propositional Logic in Proofs 4.3 The Use of Quantifiers in Proofs Counterexamples Some Theorems Involving Quantifiers 4.4 The Use of Equations in Proofs Doing the Same Thing to Both Sides of an Equation Reversibility 4.5 Mathematical Induction Axioms for the Natural Numbers The Meaning of Mathematical Induction The Structure of Proofs by Mathematical Induction Mathematical Discovery Revisited 4.6 Hints for Finding Proofs Gaining Insight into a Proof Unit 2 Sets,Relations, and Functions Chapter 5 Sets 5.1 Naive Set Theory and Russell’s Paradox Naive Set Theory The Paradoxes of Set Theory 5.2 Basic Set Operations Subsets, Proper and Otherwise The Sum Rule for Counting 5.3 More Advanced Set Operations Indexed Families of Sets Unions and Intersections of Collections of Sets Chapter 6 Relations 6.1 Ordered Pairs, Cartesian Products, and Relations Relations Inverse Relations 6.2 Equivalence Relations *6.3 Ordering Relations Preorderings Irreflexive Orderings Chapter 7 Functions 7.1 Functions and Function Notation Function Notation Why Codomains? Other Ways of Defining Functions 7.2 One-to-One and "Onto" Functions; Inverse Functions and Compositions Compositions Inverse Functions Restricting the Domain of Functions 7.3 Proofs Involving Functions Guidelines for Proving Things about Functions Induced Set Operations Inverse Images 7.4 Sequences and Inductive Definitions Definitions by Induction Justification of Inductive Definitions 7.5 Cardinality Finite and Infinite Sets Countable and Uncountable Sets 7.6 Counting and Combinatorics Permutations Combinations 7.7 The Axiom of Choice and the Continuum Hypothesis The Axiom of Choice The Continuum Hypothesis Unit 3 Number Systems Chapter 8 The Integers and the Rational Numbers 8.1 The Ring Z and the Field Q The Field of Rational Numbers 8.2 Introduction to Number Theory Complete Induction and the Fundamental Theorem of Arithmetic Some Well-Known Theorems of Number Theory Some Famous Conjectures of Number Theory *8.3 More Examples of Rings and Fields Modular Arithmetic “Well-Definedness” of Operations on Quotient Structures Finite Fields *8.4 Isomorphisms Chapter 9 The Real Number System 9.1 The Completeness Axiom Completeness of the Real Number System 9.2 Limits of Sequences and Sums of Series Infinite Series and Decimals 9.3 Limits of Functions and Continuity Continuity *9.4 Topology of the Real Line Connected Sets *9.5 The Construction of the Real Numbers The Construction of the Integers and the Rationals Chapter 10 The Complex Number System 10.1 Complex Numbers 10.2 Additional Algebraic Properties of C Appendices Appendix 1 A General-Purpose Axiom System for Mathematics Rules of Inference! Axioms Footnotes Appendix 2 Elementary Results About Fields and Ordered Fields The Field Axioms Proofs from the Field Axioms Proofs Using the Ordered Field Axioms Appendix 3 Some of the More Useful Tautologies Solutions and Hints to Selected Exercises Index Errata References List of Symbols and Notation Index