Название: Proofs and Ideas : A Prelude to Advanced Mathematics
Автор: B. Sethuraman
Издательство: MAA Press/American Mathematical Society
Год: 2022
Страниц: 350
Язык: английский
Формат: pdf (true)
Размер: 20.0 MB

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.

Abstract mathematics is a mixture of rigor and intuition, and to develop either of these, we need repeated exposure and keen practice. As students progress through high school and even early college, they are typically made to spend years working exclusively with algorithmic mathematics (“differentiate this,” “move this term over there,” “plug that value into it,”) before they are allowed to see any abstraction. It is therefore natural for students to have trouble in their first course with abstraction. They may not have seen the language of advanced mathematics. They may not yet be familiar with basic concepts. To them, even the simplest abstract ideas with which the subject works may feel overwhelming.

To help students make the change to abstract mathematics from a more algorithmic approach, many universities offer transition courses designed to provide some warm-up to the students. This book was written for one such course at California State University Northridge.

In the competition between developing the language and tools for writing proofs - statements, logical expressions, sets, functions, induction - on the one hand and developing mathematical intuition on the other, the balance in this book tips slightly towards the intuition side of the equation. I feel that many tools are best developed in context. So, while this book definitely has whole chapters on statements and on sets and functions and on induction, complete with exercises, the bulk of the book focuses on mathematical ideas and on developing creativity. Thus, the book has chapters on combinatorics (counting, the pigeonhole principle), on elementary number theory (divisibility, primes, the Unique Prime Factorization theorem and consequences), on analysis (convergence, continuity, the completeness of the real number system and consequences), on graph theory (the Konigsberg Bridge problem, Eulerian trails and circuits), and on elementary group theory (permutations, cyclic groups, matrix groups). Each of these chapters is liberally augmented with exercises, parenthetical remarks, comments on mathematical culture and tips on how to study mathematics.