Foundations of Higher Mathematics / Edition 2

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VG, no DJ, professional review copy, label on front cover, otherwise almost like new. Aqua paper covered boards and spine, second edition, first printing, hardcover. 292 + xii ... pp incl table of contents, preface, and an index. 7.5x9.5 inches. Read more Show Less

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More About This Textbook

Overview

This text introduces students to basic techniques of writing proofs and acquaints them with some fundamental ideas. The authors assume that students using this text have already taken courses in which they developed the skill of using results and arguments that others have conceived. This text picks up where the others left off — it develops the students' ability to think mathematically and to distinguish mathematical thinking from wishful thinking.
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Product Details

  • ISBN-13: 9780534929619
  • Publisher: Brooks/Cole
  • Publication date: 1/28/1992
  • Edition description: Older Edition
  • Edition number: 2
  • Pages: 320

Meet the Author

Ph.D., University of North Carolina

Ph.D., University of Georgia

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Table of Contents

Preface
1 The Logic and Language of Proofs 1
1.1 Propositions 1
1.2 Expressions and Tautologies 8
1.3 Quantifiers 16
1.4 Methods of Proof 23
1.5 The Contradiction Method of Proof 28
1.6 More Proofs 31
2 Sets 37
2.1 Introduction 37
2.2 Operations on Sets 41
2.3 Indexed Families 49
2.4 An Axiomatic Approach to Sets 54
3 Mathematical Induction 59
3.1 Proof by Induction 60
3.2 Other Principles of Induction 68
3.3 Induction and Recursion 76
3.4 Consequences of the Division Algorithm 85
3.5 Number Theory 89
4 Combinatorial Proofs 95
4.1 The Sum and Product Rules 95
4.2 Dirichlet's Pigeonhole Principle 101
4.3 The Binomial Theorem 104
4.4 Graphs 112
5 Relations and Orders 117
5.1 Relations 117
5.2 Cartesian Graphs and Directed Graphs 122
5.3 Equivalence Relations 130
5.4 Partitions and Identifications 136
5.5 Congruence 144
5.6 Composition of Relations 152
5.7 Types of Orders 155
6 Functions 161
6.1 Functions as Relations 161
6.2 Functions Viewed Globally 169
6.3 Permutations 174
6.4 Functions and Partitions 178
6.5 Real-valued Functions 180
6.6 Images and Inverse Images of Sets 188
6.7 Functions and Indexed Families 195
7 Countable and Uncountable Sets 199
7.1 Finite and Infinite Sets 201
7.2 The Schroeder-Bernstein Theorem 204
7.3 The Well-Ordering Principle and the Axiom of Choice 207
7.4 Countable Sets 210
7.5 Uncountable Sets 213
7.6 Any Infinite Set Contains a Countably Infinite Subset 217
7.7 Cardinal Arithmetic 219
8 Introduction to Groups 225
8.1 Operations 225
8.2 Integers Modulo n 229
8.3 Groups 231
8.4 Permutation Groups 237
8.5 Subgroups 239
8.6 Homomorphisms and Isomorphisms 244
8.7 Cyclic Groups 249
9 Numbers 255
9.1 The Integers 256
9.2 Order Relations on [actual symbols not reproducible] 258
9.3 Rational Numbers 261
9.4 Real Numbers 267
9.5 Complex Numbers 276
References 282
Selected Answers and Hint 283
Index 288
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  • Anonymous

    Posted March 8, 2007

    A great start for students reaching out for math

    This book was extremely well written. I was pleased to see several isolated (but important) topics developed in a coherent logical manner. Depending on what mathematical field you decide to go into, this book has at least one chapter of advanced abstractions that you will use. For example: Congruences are used heavily in Number Theory and numerical analysis, Relations and Sets are useful in nearly every field of mathematics, and of course, they have a chapter 'Introduction to Groups' which can be used as a spring board for abstract algebra. Most important, this book prepares students for what advanced mathematics really is: Proofs. For those of you who dream of getting a Ph.D. in mathematics, this book represents the 'make-or-break' class for math majors. And due to the well-written style, it makes that jump oh so easy.

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