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Primer of Abstract Mathematics (Classroom Resource Materials Series) / Edition 1

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Brand new. We distribute directly for the publisher. This book will help prepare the reader to cope with abstract mathematics, specifically abstract algebra. The intended ... audience consists of prospective math majors, those taking or intending to take a first course in abstract algebra who feel the need to strengthen their background, and students in applied fields who need some experience in dealing with abstract mathematical ideas.Learning any area of abstract mathematics will involve writing formal proofs, but it is at least as important to learn to think intuitively about the subject and to express ideas clearly and cogently using ordinary English. The author aids intuition by keeping proofs short and as informal as possible, using concrete examples which illustrate all the features of the general case, and by giving heuristic arguments when a formal development would take too long. The text can serve as a possible model on how to write mathematics for an audience with limited experience in formalism and abstraction.Ash presents several expository innovations. He presents an entirely informal development of set theory that gives students the basic results that they will need in algebra. One of the chapters which presents the theory of linear operators, introduces the Jordan Canonical Form right at the beginning, with a proof of existence at the end of the chapter. Read more Show Less

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Overview

A Primer of Abstract Mathematics prepares the reader to cope with abstract mathematics, specifically abstract algebra. It can serve as a text for prospective mathematics majors, as well as for those students taking or preparing to take a first course in abstract algebra, or those in applied fields who need experience in dealing with abstract mathematical ideas.

Learning any area of abstract mathematics involves writing formal proofs, but it is equally important to think intuitively about the subject and to express ideas clearly and cogently. The author aids intuition by keeping proofs short and as informal as possible, using concrete examples which illustrate all the features of the general case, and by giving heuristic arguments when a formal development would take too long. The text can serce as a model on how to write mathematics for an audience with limited experience in formalism and abstraction.

Ash introduces several expository innovations in A Primer of Abstract Mathematics. He presents an entirely informal development of set theory that gives students the basic results that they will need in algebra. The chapter which presents the theory of linear operators introduces the Jordan canonical Form right at the beginning, with a proof of existence at the end of the chapter.

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Editorial Reviews

Booknews
The author focuses on intuitive development of the subject and provides simple, informal, or heuristic proofs and uses concrete examples. He informally develops the set theory needed, and presents other subjects such as logic, counting, elementary number theory, linear algebra, ant linear operators. Entire solutions to the many problems in each chapter section are included. Annotation c. by Book News, Inc., Portland, Or.
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Product Details

  • ISBN-13: 9780883857083
  • Publisher: Mathematical Association of America
  • Publication date: 7/1/1998
  • Series: Classroom Resource Materials Series
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 188
  • Product dimensions: 7.02 (w) x 9.96 (h) x 0.39 (d)

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Preface
The purpose of this book is to prepare you to cope with abstract mathematics. The intended audience consists of: prospective math majors; those taking or intending to take a first course in abstract algebra who feel the need to strengthen their background; and graduate students (and possibly some undergraduates) in applied fields who need some experience in dealing with abstract mathematical ideas. If you have studied calculus, you have had some practice working with common functions and doing computations. If you have taken further courses with an applied flavor, such as differential equations and matrix algebra, you have probably begun to appreciate mathematical structure and reasoning. If you have taken a course in discrete mathematics,  you may have some experience in writing proofs. How much of this is sufficient background for the present text? I don't know; it will depend on the individual student. My suggestion would be that if you have taken some math courses, enjoyed them and done well, give it a try.
 Upon completing this book, you should be ready to handle a first course in abstract algebra. (It is also useful to prepare for a first course in abstract analysis, and one possible sources is Real Variables With basic Metric Space Topology by Robert B. Ash, IEEE Press, 1993. This basic analysis text covers the course itself as well as the preparation.)
 In studying any area of mathematics, there are, in my view, three essential factors, in order of importance:
1. Learning to think intuitively about the subject;
2. Expressing ideas clearly and cogently using ordinary English;
3. Writing formal proofs
Abstract language is used by mathematicians for precision and economy in statements and proofs, so it is certainly involved in item 3 above. But abstraction can interfere with the learning process, at all levels, so for best results in items 1 and 2, we should use abstract language sparingly. We are pulled in opposite directions and must compromise. I will try to be as informal as I can, but at some point we must confront the beast (i.e., an abstract theorem and its proof). I think you'll find that if you understand the intuition behind a mathematical statement or argument, you will have a much easier time finding your way through it.
 I've attempted to come up with a selection of topics that will help make you very comfortable when you begin to study abstract algebra. Here is a summary:
1. Logic and Foundations.  Basic logic and standard methods of proof; sets, functions and relations, especially partial ordering and equivalence relations.
2. Counting. Finite sets and standard methods of counting (permutations and combinations); countable and uncountable sets; proof that the rational numbers are countable but the real numbers are uncountable.
3. Elementary Number Theory. Some basic properties of the integers, including the Euclidean algorithm, congruence modulo m, simple diophantine equations, the Euler ? function, and the Möbius Inversion Formula.
4. Some Highly Informal Set Theory. Cardinal numbers and their arithmetic; well-ordering and its applications, including Zorn's Lemma.
5. Linear Algebra. Finite-dimensional vector spaces, along with linear transformations and their representation by matrices.
6. Theory of Linear Operators. Jordan Canonical form; minimal and characteristic polynomials; adjoints; normal operators.

A single chapter on a subject such as number theory does not replace a full course, and if  you find a particular subject interesting, I would urge you to pursue the area further. The more mathematics you study, the more skillful you will become at it.
 Another purpose of the book is to provide one possible model for how to write mathematics for an audience with limited experience in formalism and abstraction. I try to keep proofs short and as informal as possible, and to use concrete examples which illustrate all the features of the general case. When a formal development would take too long (notable in set theory), I try to replace the sequence of abstract definitions and theorems by a consistent thought process. This makes it possible to give an intuitive development of some major results. In the last chapter on linear operators, you are given a powerful engine, the Jordan Canonical Form. The proof of existence is difficult and should probably be skipped on first reading. But using the Jordan form right from the start simplifies the development considerably, and this should contribute to your understanding of linear algebra.
 Each section has a moderate number of exercises, with solutions given at the end of the book. Doing most of them will help you master the material, without (I hope) consuming too much time.
 The book may be used as a text for a course in learning how to think mathematically. The duration of the course (one semester, one quarter, two quarters) will depend on the background of the students. Chapter 3, Chapter 4, and Chapters 5-6 are almost independent. (Before studying Chapter 5, it is probably useful to look at the description of various algebraic structures at the beginning of Section 3.3 and the definition of a vector space at the end of Section 4.2.) A shorter course can be constructed by choosing one or two of these options after covering Chapters 1 and 2.
 We are doing theoretical, abstract mathematics, and students in applied fields may wonder where the applications are. But a computer scientist needs to know some elementary number theory in order to understand public key cryptography. An electrical engineer might want to study basic set theory in order to cope with abstract algebra and thereby learn about error-correcting codes. A statistician needs to know some theoretical linear algebra (projections, diagonalization of symmetric matrices, quadratic forms) in order to work with the multivariate normal distribution. There is potentially a large audience for abstract mathematics, and to reach this audience it is not necessary for us to teach detailed physical and engineering applications. The physics and engineering departments are quite capable of doing this. It is certainly useful to suggest possible applications, and as an illustration, I have included an appendix giving a typical application of linear algebra. But it is essential that we written in an accessible and congenial style, and give informal or heuristic arguments when appropriate.

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Table of Contents


Chapter 1 Logic and Foundations
1.1 Truth Tables
1.2 Quantifiers
1.3 Proofs
1.4 Sets
1.5 Functions
1.6 Relations

Chapter 2 Counting
2.1 Fundamentals
2.2 The binomial and Multinomial Theorems
2.3 The Principle of Inclusion and Exclusion
2.4 Counting Infinite Sts

Chapter 3 Elementary Number Theory
3.1 The Euclidean Algorithm
3.2 Unique Factorizations
3.3 Algebraic Structures
3.4 Further Properties of Congruence Modulo m
3.5 Linear Diophantine Equations and Simultaneous Congruences
3.6 Theorems of Euler and Fermat
3.7 The Möbius Inversion Formula

Chapter 4 Some Highly Informal Set Theory
4.1 Well-Orderings
4.2 Zorn's Lemma and the Axiom of Choice
4.3 Cardinal Numbers
4.4 Addition and Multiplication of Cardinals

Chapter 5 Linear Algebra
5.1 Matrices
5.2 Determinations and Inverses
5.3 The Vector Space Fn; Linear Independence and Bases
5.4 Subspaces
5.5 Linear Transformations
5.6 Inner Product Spaces
5.7 Eigenvalues and Eigenvectors

Chapter 6 Theory of Linear Operators
6.1 Jordan Canonical Form
6.2 The Minimal and Characteristic Polynomials
6.3 The Adjoint of a Linear Operator
6.4 Normal Operators
6.5 The Existence of the Jordan Canonical Form
Appendix: An Application of Linear Algebra
Solutions to Problems
List of Symbols
Index

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