This is at heart a fairly conventional transition text, but it has a number of features that encourage students to practice and improve their mathematical reading, writing, and proving skills. I doubt that it’s possible for students to learn these skills only from a book: they need a careful reader to critique their work, so this book needs to be used with a teacher or other knowledgeable critic and not for individual study. The first edition appeared (and was reviewed) in 2003; this second edition has the same approach and outline, with a modest amount of additional material, and twice as many problems as before.
The book is primarily concerned with an exposition of those parts of mathematics in which students need a more thorough grounding before they can work successfully in upper-division undergraduate courses. This means that most of the book deals with logic and set theory, with some study of the real numbers and completeness. Layered on top of this conventional material are a number of features to encourage the students to build their skills in reading, writing, and proving. The book leads by example: the proofs that are given are written out fully and carefully, and are intended to be a model the students can use to construct their own proofs. The remaining material, which comprises most of the book, is presented in the form of exercises (easy, and solved in the text) and problems (harder, and including some proofs). There are several “nontheorems” and “not-a-proofs” that are fallacious and whose fallacies the student is intended to uncover.
The book ends with a series of longer projects from several areas of mathematics (not just those covered in the body of the work), that the student is supposed to figure out and write up carefully. These are very structured, in the sense that they are already broken down into a series of simpler steps. If you think (as I do) that this simplifies the task too much, you may want to supplement this text with more open-ended projects, such as those in Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go by Crannell et al.
Bottom line: a mathematically-conventional but pedagogically-innovative take on transition courses.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.
-Preface. -1. The How, When, and Why of Mathematics.- 2. Logically Speaking.- 3.Introducing the Contrapositive and Converse.- 4. Set Notation and Quantifiers.- 5. Proof Techniques.- 6. Sets.- 7. Operations on Sets.- 8. More on Operations on Sets.- 9. The Power Set and the Cartesian Product.- 10. Relations.- 11. Partitions.- 12. Order in the Reals.- 13. Consequences of the Completeness of (\Bbb R).- 14. Functions, Domain, and Range.- 15. Functions, One-to-One, and Onto.- 16. Inverses.- 17. Images and Inverse Images.- 18. Mathematical Induction.- 19. Sequences.- 20. Convergence of Sequences of Real Numbers.- 21. Equivalent Sets.- 22. Finite Sets and an Infinite Set.- 23. Countable and Uncountable Sets.- 24. The Cantor-Schröder-Bernstein Theorem.- 25. Metric Spaces.- 26. Getting to Know Open and Closed Sets.- 27. Modular Arithmetic.- 28. Fermat’s Little Theorem.- 29. Projects.- Appendix.- References.- Index.