Sách BRIDGE TO ABSTRACT MATHEMATICS Mathematical Proof and Structures - Ronald E Morash

BRIDGE TO ABSTRACT MATHEMATICS Mathematical Proof and Structures - Ronald E Morash

Contents
BOOK ONE The Foundation: Sets, Logic, and Mathematical Argument
SETS 3
Basic Definitions and Notation 4
Operations on Sets 15
Algebraic Properties of Sets 29
Theorems of Set Theory 39
Counting Properties of Finite Sets (Optional) 42
LOGIC, PART I: THE PROPOSITIONAL CALCULUS 52
Basic Concepts of the Propositional Calculus 53
Tautology, Equivalence, the Conditional, and Biconditional 59
Theorems of the Propositional Calculus 64
Analysis of Arguments for Logical Validity, Part I (Optional) 76
LOGIC, PART 11: THE PREDICATE CALCULUS 81
Basic Concepts of the Predicate Calculus 82
Quantification 88
Theorems About Predicates in One Variable 95
Quantification of Propositional Functions in Several Variables 101
Analysis of Arguments for Logical Validity, Part I1 (Optional) 110
ELEMENTARY 'APPLICATIONS OF LOGIC 115
Applications of Logic to Set Theory-Some Proofs 115
Infinite Unions and Intersections 124
The Limit Concept (Optional) 128
METHODS OF MATHEMATICAL PROOF,
PART 1: ELEMENTARY METHODS 146
Conclusions Involving V, but Not 3 or +. Proof by Transitivity 148
Conclusions Involving V and +, but Not 3 160
Proof by Specialization and Division into Cases 170
Proof by Mathematical Induction 178
METHODS OF MATHEMATICAL PROOF,
PART 11: ADVANCED METHODS 190
Conclusions Involving V, followed by 3 (Epsilon-Delta Proofs Optional) 190
Indirect Proofs 204
Existence and Uniqueness (Optional) 212
Preview of Additional Advanced Methods of Proof (Optional) 222
BOOK TWO Bridging Topics: Relations, Functions,
and Number Systems
RELATIONS, PART I: EQUIVALENCE RELATIONS
AND PARTIAL ORDERINGS 227
Relations 228
Equivalence Relations 235
Equivalence Classes and Partitions 240
Partial Ordering 245
RELATIONS, PART Ik FUNCTIONS
AND MAPPINGS 252
Functions and Mappings 252
More on Functions and Mappings-Surjections, Bijections, Image,
and Inverse Image 266
Cardinal Number of a Set 277
Arbitrary Collections of Sets 289
PROPERTIES OF THE NUMBER SYSTEMS
OF UNDERGRADUATE MATHEMATICS 293
Fields 294
Ordered Fields 303
Completeness in an Ordered Field 310
Properties of the Complex Number Field 319
CONSTRUCTION OF THE NUMBER SYSTEMS
OF UNDERGRADUATE MATHEMATICS 329
An Axiomatization for the System of Positive Integers 330
Development of the Integers and Rational Numbers 343
Outline of a Construction of the Reds 352
Answers and Solutions to Selected Exercises 362
List of Symbols 380
Index 383