Enhanced descriptions from Syndetics:

This book eases students into the rigors of university mathematics. The emphasis is on understanding and constructing proofs and writing clear mathematics. The author achieves this by exploring set theory, combinatorics, and number theory, topics that include many fundamental ideas and may not be a part of a young mathematician's toolkit. 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 all-time-great 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. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas.

Table of contents provided by Syndetics

• 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

Reviews provided by Syndetics

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