Write Your Own Proofs. Amy BabichЧитать онлайн книгу.

      Foreword A

      This Book and How to Use It

      About twenty years ago, American universities began to offer a mathematics course (called variously Set Theory and Logic or Discrete Mathematics, at different institutions) designed to serve as an introduction to mathematical proof. The usual assigned textbook for this course is either Chapter Zero of a graduate mathematics text, or one of the recent textbooks on Discrete Mathematics (by Grimaldi, Epp, Rosen, et al.). Neither choice of textbook is ideal for the novice professor assigned to teach this course. Chapter Zero of a graduate textbook is mere outline, with very few exercises, and the Discrete Mathematics books are all much too large for a one-semester course. Nearly always, the rookie instructor for this course must make her/his own class notes and supplementary problem sets. This takes time and effort, and the first-time instructor is often short of both.

      Like other instructors of this course, we made our own class notes. We also traded our notes back and forth, making use of each other’s ideas. We used the first half of Introduction to Set Theory and Logic by Lin and Lin, as a model, but we changed it to suit our purposes. We did not find that the section on deductive reasoning (featuring syllogisms, Modus Ponens, and Modus Tollens) helped our students to write proofs, so we jettisoned it. There is a great deal of material in this sort of course, and we move through it slowly, at the students’ pace. We omit whatever we don’t find helpful.

      We used these class notes for several semesters, changing them as we went along. The main change that occurred with time and use was the addition of examples and exercises in sections that the students found difficult. The notes, even in their most rudimentary form, were noticeably more effective than our old textbooks for helping students to learn. Students said that they liked the notes, but wished that they were a real book.

      So we set about turning the notes into a book. (This turned out to be a lot of trouble.) We tried to write down the explanations we gave orally in class. These appear as ^πRemarks” in the text. The Remarks are there to help whoever may find them helpful. If a Remark seems unhelpful or confusing, feel free to ignore it. The Remarks are optional aids to understanding.

      We have been at some pains not to include all possible material in this text. When writing a book of this sort, one quickly comes to understand why the available books (by Epp, Grimaldi, Rosen, et al.) are so huge. The material here has very natural connections to number theory, probability, graph theory, complexity theory, transfinite arithmetic, and so on. But we do not wish to include an introductory course in each of these subjects, although