0.5: Advice to the Instructor
Learning terminology - what do "contrapositive" and "converse" mean - comes easily to most students. Your challenge in the course is to teach them how to read definitions closely, and then how to manipulate them. This is much harder when there is no concrete image that students can keep in mind. Vectors in \(\mathbb{R}^{n}\) , for example, are more intimidating than in \(\mathbb{R}^{3}\) , not because of any great inherent increase in complexity, but because they are harder to think of geometrically, so students must trust the algebra alone. This trust takes time to build.
Chapter 1 is mainly to establish notation and discuss necessary concepts that some may have already seen (like injections and surjections). Unfortunately this may be the first exposure to some of these ideas for many students, so the treatment is rather lengthy. The speed at which the material is covered naturally will depend on the strength and background of the students. Take some time explaining why a sequence can be thought of as a function with domain \(\mathbb{N}\) - variations on this idea will recur.
Chapter 2 introduces relations. These are hard to grasp, because of the abstract nature of the definition. Equivalences and linear orderings recur throughout the book, and students’ comfort with these will increase.
Neither Chapter 1 nor Chapter 2 dwell on proofs. In fact mathematical proofs and elementary first order logic are not introduced until Chapter 3. Our objective is to get the student thinking about mathematical structures and definitions without the additional psychic weight of reading and writing proofs. We use examples to illustrate the definitions. The first Chapters provide basic conceptual foundations for later chapters, and we find that most students have their hands full just trying to read and understand the definitions and examples. In the exercises we ask the students to "show" the truth of some mathematical claims. Our intention is to get the student thinking about the task of proving mathematical claims. It is not expected that they will write successful arguments before Chapter 3 . We encourage the students to attempt the problems even though they will likely be uncertain about the requirements for a mathematical proof. If you feel strongly that mathematical proofs need to be discussed before launching into mathematical definitions, you can cover Chapter 3 first.
Chapter 3 is fairly formal, and should go quickly. Chapter 4 introduces students to the first major proof technique - induction. With practice, they can be expected to master this technique. We also introduce as an ongoing theme the study of polynomials, and prove for example that a polynomial has no more roots than its degree.
Chapters 5,6 and 7 are completely independent of each other. Chapter 5 treats limits and continuity, up to proving that the uniform limit of a sequence of continuous functions is continuous. Chapter 6 is on infinite sets, proving Cantor’s theorems and the Schröder-Bernstein theorem. By the end of the chapter, the students will have come to appreciate that it is generally much easier to construct two injections than one bijection!
Chapter 7 contains a little number theory - up to the proof of Fermat’s little theorem. It then shows how much of the structure transfers to the algebra of real polynomials.
Chapter 8 constructs the real numbers, using Dedekind cuts, and proves that they have the least upper bound property. This is then used to prove the basic theorems of real analysis - the Intermediate Value theorem and the Extreme Value theorem. Sections \(8.1\) through \(8.4\) require only Chapters \(1-4\) and Section 6.1. Sections \(8.5-8.8\) require Sections \(5.1\) and 5.2. Section \(8.9\) requires Chapter 6 .
In Chapter 9, we introduce the complex numbers. Sections \(9.1\) \(9.3\) prove the Tartaglia-Cardano formula for finding the roots of a cubic, and point out how it is necessary to use complex numbers even to find real roots of real cubics. These sections require only Chapters 1 - 4. In Section \(9.4\) we prove the Fundamental Theorem of Algebra. This requires Chapter 5 and the Bolzano-Weierstrass theorem from Section 8.6.
What is a reasonable course based on this book? Chapters 1 - 4 are essential for any course. In a one quarter course, one could also cover Chapter 6 and either Chapter 5 or 7 . In a semester-long course, one could cover Chapters \(1-6\) and one of the remaining three chapters. Chapter 9 can be covered without Chapter 8 if one is willing to assert the Least Upper Bound property as an axiom of the real numbers, and then Section \(8.6\) can be covered before Section \(9.4\) without any other material from Chapter 8 .
We suggest that you agree with your colleagues on a common curriculum for this course, so that topics that you cover thoroughly (e.g. cardinality) need not be repeated in successive courses.
This transition course is becoming one of the most important courses in the mathematics curriculum, and the first important course for the mathematics major. For the talented and intellectually discriminating first or second year student the standard early courses in the mathematics curriculum - calculus, differential equations, matrix algebra - provide little incentive for studying mathematics. Indeed, there is little mathematics in these courses, and less still with the evolution of lower undergraduate curricula towards the service of the sciences and engineering. This is particularly disturbing as it pertains to the talented student who has not yet decided on a major and may never have considered mathematics. We believe that the best students should be encouraged to take this course as early as possible - even concurrent with the second semester or third quarter of first year calculus. It is not just to help future math majors, but can also serve a valuable rôle in recruiting them, by letting smart students see that mathematics is challenging and, more to the point, interesting and deep. Mathematics is its own best apologist. Expose the students early to authentic mathematical thinking and results and let them make an informed choice. It may come as a surprise to some, but good students still seek what mathematicians sought as students - the satisfaction of mastering a difficult, interesting and useful discipline.