Higher Education

A Transition to Advanced Mathematics, 7th Edition

  • Doug Smith University of North Carolina at Wilmington
  • Maurice Eggen Trinity University
  • Richard St. Andre Central Michigan University
  • ISBN-10: 0495562025  |  ISBN-13: 9780495562023
  • 416 Pages
  • Previous Editions: 2006, 2001, 1997
  • © 2011 | Published
  • College Bookstore Wholesale Price = $228.75
  • Newer Edition Available
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A TRANSITION TO ADVANCED MATHEMATICS helps students make the transition from calculus to more proofs-oriented mathematical study. The most successful text of its kind, the 7th edition continues to provide a firm foundation in major concepts needed for continued study and guides students to think and express themselves mathematically—to analyze a situation, extract pertinent facts, and draw appropriate conclusions. The authors place continuous emphasis throughout on improving students' ability to read and write proofs, and on developing their critical awareness for spotting common errors in proofs. Concepts are clearly explained and supported with detailed examples, while abundant and diverse exercises provide thorough practice on both routine and more challenging problems. Students will come away with a solid intuition for the types of mathematical reasoning they'll need to apply in later courses and a better understanding of how mathematicians of all kinds approach and solve problems.

Features and Benefits

  • The authors follow a logical development of topics, and write in a readable style that is consistent and concise. As each new mathematical concept is introduced the emphasis remains on improving students' ability to write proofs.
  • Worked examples and exercises throughout the text, ranging from the routine to the challenging, reinforce the concepts.
  • Proofs to Grade exercises test students' ability to distinguish correct reasoning from logical or conceptual errors.
  • A flexible organization allows instructors to expand coverage or emphasis on certain topics and include a number of optional topics without any disruption to the flow or completeness of the core material.

Table of Contents

Propositions and Connectives. Conditionals and Biconditionals. Quantifiers. Basic Proof Methods I. Basic Proof Methods II. Proofs Involving Quantifiers. Additional Examples of Proofs
Basic Notions of Set Theory. Set Operations. Extended Set Operations and Indexed Families of Sets. Induction. Equivalent Forms of Induction. Principles of Counting.
Relations. Equivalence Relations. Partitions. Ordering Relations. Graphs.
Functions as Relations. Constructions of Functions. Functions That Are Onto; One-to-One Functions. One-to-One Correspondences and Inverse Functions. Images of Sets. Sequences.
Equivalent Sets; Finite Sets. Infinite Sets. Countable Sets. The Ordering of Cardinal Numbers. Comparability of Cardinal Numbers and the Axiom of Choice.
Algebraic Structures. Groups. Subgroups. Operation Preserving Maps. Rings and Fields.
Ordered Field Properties of the Real Numbers. The Heine-Borel Theorem. The Bolzano-Weierstrass Theorem. The Bounded Monotone Sequence Theorem. Comparability of Cardinals and the Axiom of Choice.

What's New

  • An expanded Preface to the Student reviews what students are expected to know about standard number systems, sets, functions, and basic terminology at the beginning of the course, outlines why the ability to do proofs is important, and provides practice exercises verifying properties of integers, rational numbers, and real numbers.
  • Several key sections crucial to student understanding have been extensively revised for improved clarity. Chapter 1 (Logic and Proofs) now closes with a summary of proof methods, reinforced with revamped examples and exercises. Results from elementary number theory in the first two chapters are now better highlighted and tied together more cohesively. Chapter 4 (Functions) is reorganized for improved flow, with review of the most basic properties and terminology moved to the Preface to the Student and study of additional properties of functions now included. The section on countable sets in Chapter 5 has been completely rewritten for greater accessibility.
  • Exercise sets have been thoroughly refreshed, with new exercises in almost every section and expanded Proofs to Grade exercises.
  • Throughout the text biographical footnotes have been added to provide a historical perspective on the development of the foundations of mathematics.


All supplements have been updated in coordination with the main title. Select the main title's "About" tab, then select "What's New" for updates specific to title's edition.

For more information about these supplements, or to obtain them, contact your Learning Consultant.

Instructor Supplements

Solution Builder  (ISBN-10: 0495826693 | ISBN-13: 9780495826699)

Offers fully worked instructor solutions to all exercises in the text in customizable online format. Adopting instructors can sign up for access at www.cengage.com/solutionbuilder.

Meet the Author

Author Bio

Doug Smith

The authors are the leaders in this course area. They decided to write this text based upon a successful transition course that Richard St. Andre developed at Central Michigan University in the early 1980s. This was the first text on the market for a transition to advanced mathematics course and it has remained at the top as the leading text in the market. Douglas Smith is Professor of Mathematics at the University of North Carolina at Wilmington. Dr. Smith’s fields of interest include Combinatorics / Design Theory (Team Tournaments, Latin Squares, and applications), Mathematical Logic, Set Theory, and Collegiate Mathematics Education.

Maurice Eggen

Maurice Eggen is Professor of Computer Science at Trinity University. Dr. Eggen's research areas include Parallel and Distributed Processing, Numerical Methods, Algorithm Design, and Functional Programming.

Richard St. Andre

Richard St. Andre is Associate Dean of the College of Science and Technology at Central Michigan University. Dr. St. Andre's teaching interests are quite diverse with a particular interest in lower division service courses in both mathematics and computer science.