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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000010144586 | QA9.54 E22 1997 | Open Access Book | Book | Searching... |
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Summary
Summary
The purpose of this book is to introduce the basic ideas of mathematical proof to students embarking on university mathematics. The emphasis is on helping the reader in understanding and constructing proofs and writing clear mathematics. This is achieved by exploring set theory, combinatorics and number theory, topics which include many fundamental ideas which are part of the tool kit of any mathematician. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. Over 250 problems include questions to interest and challenge the most able student as well as plenty of routine exercises to help familiarize the reader with the basic ideas.
Reviews 1
Choice Review
Eccles writes to introduce the basic ideas of mathematical proof to students embarking on a study of university mathematics in the British University setting; he has aimed the book at first-year honors students in mathematics. American readers may recognize this as an appropriate text for use in the "stepping stone" course that often is placed in the sophomore year between the study of calculus and the rigors of upper-division mathematical course work. Such a course primarily aims to lead the student away from the notion that mathematics is synonymous with computation, to acquaint the student with the language and symbology of mathematics, and to emphasize the skills necessary to read and write mathematical proofs. Though Eccles offers the obligatory section on proof techniques, it is mercifully brief, as he seems to realize that actually doing proofs is a more effective pedagogical tool than talking about them. The student learns about proof techniques by being presented with a rigorous study of several fundamental topics pervasive in mathematics, including sets, functions, cardinality, combinatorics, and modular arithmetic. A student planning to study advanced mathematics would be well served by first mastering the material in this book. Lower-division undergraduates. D. S. Larson Gonzaga University
Table of Contents
| Part I Mathematical Statements and Proofs |
| 1 The language of mathematics |
| 2 Implications |
| 3 Proofs |
| 4 Proof by contradiction |
| 5 The induction principle |
| Part II Sets and Functions |
| 6 The language of set theory |
| 7 Quantifiers |
| 8 Functions |
| 9 Injections, surjections and bijections |
| Part III Numbers and Counting |
| 10 Counting |
| 11 Properties of finite sets |
| 12 Counting functions and subsets |
| 13 Number systems |
| 14 Counting infinite sets |
| Part IV Arithmetic |
| 15 The division theorem |
| 16 The Euclidean algorithm |
| 17 Consequences of the Euclidean algorithm |
| 18 Linear diophantine equations |
| Part V Modular Arithmetic |
| 19 Congruences of integers |
| 20 Linear congruences |
| 21 Congruence classes and the arithmetic of remainders |
| 22 Partitions and equivalence relations |
| Part VI Prime Numbers |
| 23 The sequence of prime numbers |
| 24 Congruence modulo a prime |
| Solutions to exercises |
