**AMS/MAA Textbooks**

Volume: 68;
2021;
334 pp;
Softcover

MSC: Primary 00;

**Print ISBN: 978-1-4704-6514-8
Product Code: TEXT/68**

List Price: $85.00

AMS Member Price: $63.75

MAA Member Price: $63.75

**Electronic ISBN: 978-1-4704-6761-6
Product Code: TEXT/68.E**

List Price: $85.00

AMS Member Price: $63.75

MAA Member Price: $63.75

#### You may also like

#### Supplemental Materials

# Proofs and Ideas: A Prelude to Advanced Mathematics

Share this page
*B. Sethuraman*

MAA Press: An Imprint of the American Mathematical Society

Proofs and Ideas serves as a gentle
introduction to advanced mathematics for students who previously have
not had extensive exposure to proofs. It is intended to ease the
student's transition from algorithmic mathematics to the world of
mathematics that is built around proofs and concepts.

The spirit of the book is that the basic tools of abstract
mathematics are best developed in context and that creativity and
imagination are at the core of mathematics. So, while the book has
chapters on statements and sets and functions and induction, the bulk
of the book focuses on core mathematical ideas and on developing
intuition. Along with chapters on elementary combinatorics and
beginning number theory, this book contains introductory chapters on
real analysis, group theory, and graph theory that serve as gentle
first exposures to their respective areas. The book contains hundreds
of exercises, both routine and non-routine.

This book has been used for a transition to advanced mathematics
courses at California State University, Northridge, as well as for a
general education course on mathematical reasoning at Krea University,
India.

#### Readership

Undergraduate students interested in an introduction to proofs.

#### Table of Contents

# Table of Contents

## Proofs and Ideas: A Prelude to Advanced Mathematics

- Title page 44
- Copyright 55
- Contents 88
- Preface 1212
- Chapter 1. Introduction 1616
- Chapter 2. The Pigeonhole Principle 3232
- Chapter 3. Statements 4242
- Chapter 4. Counting, Combinations 6464
- Chapter 5. Sets and Functions 8484
- Chapter 6. Interlude: So, How to Prove It? An Essay 116116
- Chapter 7. Induction 122122
- Chapter 8. Cardinality of Sets 136136
- Chapter 9. Equivalence Relations 164164
- Chapter 10. Unique Prime Factorization in the Integers 190190
- Chapter 11. Sequences, Series, Continuity, Limits 212212
- Chapter 12. The Completeness of R 242242
- 12.1. Least Upper Bound Property (LUB) 243243
- 12.2. Greatest Lower Bound Property 247247
- 12.3. Archimedean Property 248248
- 12.4. Monotone Convergence Theorem 249249
- 12.5. Bolzano-Weierstrass Theorem 251251
- 12.6. Nested Intervals Theorem 253253
- 12.7. Cauchy sequences 256256
- 12.8. Convergence of Series 258258
- 12.9. 𝑛-th roots of positive real numbers 264264
- 12.10. Further Exercises 267267
- Notes 273273

- Chapter 13. Groups and Symmetry 276276
- 13.1. Symmetries of an equilateral triangle 277277
- 13.2. Symmetries of a square 281281
- 13.3. Symmetries of an 𝑛-element set 285285
- Groups 287287
- 13.4. Subgroups 294294
- 13.5. Cosets, Lagrange’s Theorem 297297
- 13.6. Symmetry 302302
- 13.7. Isomorphisms Between Groups 306306
- 13.8. Further Exercises 311311

- Chapter 14. Graphs: An Introduction 318318
- Index 348348