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 interest. 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 with opportunities to confirm their understanding of core material, learn new concepts, and develop mathematical creativity.
About the Author
John Meier is Professor of Mathematics at Lafayette College, Pennsylvania, 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, New York, 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, Pennsylvania. 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.
Table of Contents
1. Let's play!; 2. Discovering and presenting mathematics; 3. Sets; 4. The integers and the fundamental theorem of arithmetic; 5. Functions; 6. Relations; 7. Cardinality; 8. The real numbers; 9. Probability and randomness; 10. Algebra and symmetry; 11. Projects; Appendix A. Solutions, answers, or hints to in-text exercises; Index; Bibilography.