Bridge to Abstract Mathematics: Mathematical Proof and Structures

CONTENTS xiii

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