John Meier - Exploring Mathematics: An Engaging Introduction to Proof

Exploring Mathematics gives students experience with doing mathematics interrogating mathematical claims, exploring definitions, forming conjectures, attempting proofs, and presenting results and engages them with examples, exercises, and projects that pique their curiosity. Written with a minimal number of pre-requisites, this text can be used by college students in their first and second years of study and by independent readers who want an accessible introduction to theoretical mathematics. Core topics include proof techniques, sets, functions, relations, and cardinality, with selected additional topics that provide many possibilities for further exploration.

With a problem-based approach to investigating the material, students develop interesting examples and theorems through numerous exercises and projects. In-text exercises, with complete solutions or robust hints included in an appendix, help students explore and master the topics being presented. The end-of-chapter exercises and projects provide students opportunities to confirm their understanding of core material, learn new concepts, and develop mathematical creativity.

John Meier is the David M. 70 and Linda Roth Professor of Mathematics at Lafayette College, where he also served as Dean of the Curriculum. His research focuses on geometric group theory and involves algorithmic, combinatorial, geometric, and topological issues that arise in the study of infinite groups. In addition to teaching awards from Cornell University and Lafayette College, Professor Meier is the proud recipient of the James Crawford Teaching Prize from the Eastern Pennsylvania and Delaware section of the Mathematical Association of America.

Derek Smith is Associate Professor of Mathematics at Lafayette College. His research focuses on algebra, combinatorics, and geometry. He has taught a wide variety of undergraduate courses in mathematics and other subjects in both the United States and Europe. He is the recipient of multiple teaching awards at Lafayette, and his work has been supported by the Mathematical Association of America and the National Science Foundation. Professor Smith is a former editor of the problem section of Math Horizons .

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John Meier and Derek Smith 2017

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First published 2017

Printed in the United States of America by Sheridan Books, Inc.

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ISBN 978-1-107-12898-9 Hardback

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Mathematics is a fascinating discipline that calls for creativity, imagination, and the mastery of rigorous standards of proof. This book introduces students to these facets of the field in a problem-focused setting. For over a decade, we and many others have used draft chapters of Exploring Mathematics as the primary text for Lafayettes Transition to Theoretical Mathematics course. Our collective experience shows that this approach assists students in their transition from primarily computational classes toward more advanced mathematics, and it encourages them to continue along this path by demonstrating that while mathematics can at times be challenging, it is also very enjoyable.

Here are some of the key features of Exploring Mathematics .

These and other fundamental aspects of Exploring Mathematics are described in greater detail below, where we also indicate different ways an instructor can map out the material that can be covered in a single term.

Our experience teaching courses that introduce students to mathematical proofs shows that the spirit of mathematics is effectively taught with a focus on problem-solving. It is in doing mathematics by exploring definitions, forming conjectures, and working on the writing of proofs that students begin to understand the discipline. We have packed into this book well over 600 exercises that engage students in these important activities.