Introduction to Mathematical Proofs: A Transition / Edition 1

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Overview

Shows How to Read & Write Mathematical Proofs
Ideal Foundation for More Advanced Mathematics Courses

Introduction to
Mathematical Proofs
: A Transition facilitates a smooth transition from courses designed to develop computational skills and problem solving abilities to courses that emphasize theorem proving. It helps students develop the skills necessary to write clear, correct, and concise proofs.

Unlike similar textbooks, this one begins with logic since it is the underlying language of mathematics and the basis of reasoned arguments. The text then discusses deductive mathematical systems and the systems of natural numbers, integers, rational numbers, and real numbers. It also covers elementary topics in set theory, explores various properties of relations and functions, and proves several theorems using induction. The final chapters introduce the concept of cardinalities of sets and the concepts and proofs of real analysis and group theory. In the appendix, the author includes some basic guidelines to follow when writing proofs.

Written in a conversational style, yet maintaining the proper level of mathematical rigor, this accessible book teaches students to reason logically, read proofs critically, and write valid mathematical proofs. It will prepare them to succeed in more advanced mathematics courses,
such as abstract algebra and geometry.

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Product Details

  • ISBN-13: 9781420069556
  • Publisher: Taylor & Francis
  • Publication date: 6/24/2009
  • Series: Textbooks in Mathematics Series
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 433
  • Product dimensions: 6.40 (w) x 9.30 (h) x 1.10 (d)

Meet the Author

Charles E. Roberts, Jr. is a professor of mathematics and computer science at Indiana State University.

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Table of Contents

Preface vii

1 Logic 1

1.1 Statements, Negation, and Compound Statements 2

1.2 Truth Tables and Logical Equivalences 9

1.3 Conditional and Biconditional Statements 21

1.4 Logical Arguments 29

1.5 Open Statements and Quantifiers 45

1.6 Chapter Review 60

2 Deductive Mathematical Systems and Proofs 69

2.1 Deductive Mathematical Systems 69

2.2 Mathematical Proofs 80

2.2.1 Techniques for Proving the Conditional Statement P → Q 81

2.2.2 Additional Proof Techniques 89

2.2.3 Conjectures, Proofs, and Disproofs 100

2.2.3 The System of Rational Numbers and the System of Real Numbers 106

2.3 Chapter Review 119

3 Set Theory 125

3.1 Sets and Subsets 126

3.2 Set Operations 133

3.3 Additional Set Operations 144

3.4 Generalized Set Union and Intersection 159

3.5 Chapter Review 168

4 Relations 175

4.1 Relations 175

4.2 The Order Relations <, ≤, >, ≥ 187

4.3 Reflexive, Symmetric, Transitive, and Equivalence Relations 197

4.4 Equivalence Relations, Equivalence Classes, and Partitions 205

4.5 Chapter Review 213

5 Functions 219

5.1 Functions 219

5.2 Onto Functions, One-to-One Functions, and One-to-One Correspondences 231

5.3 Inverse of a Function 240

5.4 Images and Inverse Images of Sets 249

5.5 Chapter Review 256

6 Mathematical Induction 261

6.1 Mathematical Induction 261

6.2 The Well-Ordering Principle and the Fundamental Theorem of Arithmetic 267

7 Cardinalities of Sets 275

7.1 Finite Sets 276

7.2 Denumerable and Countable Sets 283

7.3 Uncountable Sets 289

8 Proofs from Real Analysis 297

8.1 Sequences 297

8.2 Limit Theorems for Sequences 307

8.3 Monotone Sequences and Subsequences 313

8.4 Cauchy Sequences 321

9 Proofs from Group Theory 325

9.1 Binary Operations and Algebraic Structures 325

9.2 Groups 331

9.3 Subgroups and Cyclic Groups 338

A Reading and Writing Mathematical Proofs 349

Answers to Selected Exercises 357

References 415

Index 417

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