Bridge to abstract mathematics. Mathematical Proof and Structures

Cover
Title page
Preface
BOOK ONE The Foundation: Sets, Logic, and Mathematical Argument
1 SETS
1.1 Basic Definitions and Notation
1.2 Operations on Sets
1.3 Algebraic Properties of Sets
1.4 Theorems of Set Theory
1.5 Counting Properties of Finite Sets (Optional)
2 LOGIC, PART I: THE PROPOSITIONAL CALCULUS
2.1 Basic Concepts of the Propositional Calculus
2.2 Tautology, Equivalence, the Conditional, and Biconditional
2.3 Theorems of the Propositional Calculus
2.4 Analysis of Arguments for Logical Validity, Part 1 (Optional)
3 LOGIC, PART II: THE PREDICATE CALCULUS
3.1 Basic Concepts of the Predicate Calculus
3.2 Quantification
3.3 Theorems About Predicates in One Variable
3.4 Quantification of Propositional Functions in Several Variables
3.5 Analysis of Arguments for Logical Validity, Part II (Optional)
4 ELEMENTARY APPLICATIONS OF LOGIC
4.1 Applications of Logic to Set Theory-Some Proofs
4.2 Infinite Unions and Intersections
4.3 The Limit Concept (Optional)
5 METHODS OF MATHEMATICAL PROOF, PART 1: ELEMENTARY METHODS
5.1 Conclusions Involving ∀, but Not ∃ or ->. Proof by Transitivity
5.2 Conclusions Involving ∀ and ->, but Not ∃
5.3 Proof by Specialization and Division into Cases
5.4 Proof by Mathematical Induction
6 METHODS OF MATHEMATICAL PROOF, PART II: ADVANCED METHODS
6.1 Conclusions Involving ∀, followed by ∃ (Epsilon-Delta Proofs Optional)
6.2 Indirect Proofs
6.3 Existence and Uniqueness (Optional)
6.4 Preview of Additional Advanced Methods of Proof (Optional)
BOOK TWO Bridging Topics: Relations, Functions, and Number Systems
7 RELATIONS, PART I: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS
7.1 Relations
7.2 Equivalence Relations
7.3 Equivalence Classes and Partitions
7.4 Partial Orderings
8 RELATIONS, PART II: FUNCTIONS AND MAPPINGS
8.1 Functions and Mappings
8.2 More on Functions and Mappings-Surjections, Bijections, Image, and Inverse Image
8.3 Cardinal Number of a Set
8.4 Arbitrary Collections of Sets
9 PROPERTIES OF THE NUMBER SYSTEMS OF UNDERGRADUATE MATHEMATICS
9.1 Fields
9.2 Ordered Fields
9.3 Completeness in an Ordered Field
9.4 Properties of the Complex Number Field
10 CONSTRUCTION OF THE NUMBER SYSTEMS OF UNDERGRADUATE MATHEMATICS
10.1 An Axiomatization for the System of Positive Integers
10.2 Development of the Integers and Rational Numbers
10.3 Outline of a Construction of the Reals
Answers and Solutions to Selected Exercises
List of Symbols
Index