MATH 341, Section 001 FALL 2014 Introduction to the Language and Practice of Mathematics

Transcription

1 MATH 341, Section 001 FALL 2014 Introduction to the Language and Practice of Mathematics Class Meetings: MW 9:30-10:45 am in EMS E424A, September 3 to December 10 [Thanksgiving break November 26 30; final exam December 17] Instructor: Allen Bell Contact information: EMS E adbell@uwm.edu D2L: Go to then log in and find Introduction to the Language and Practice of Mathematics (Math 341). Text and Topics: How to Prove It: A Structured Approach, second edition, by Daniel J. Velleman, Cambridge University Press. (In September 2014, the text is $26 at Amazon. You can view parts of the book at Amazon or at sxt-rollnhcc&printsec=frontcover&dq=how+to+prove+it:+a+structured+approach.) We will cover much of the book, plus possibly some material not in the book. I will also post some very concise notes that I have used in the past on the D2L site. I m not sure whether or how these will be used in our class. Course Goals and Format: The primary goal of this class is to develop your ability to read, write, communicate, and understand logical reasoning and rigorous mathematics. Successful completion of this course should serve as preparation for the more theoretical upper level mathematics courses. In the process, we will touch on some topics that are not typically emphasized in lower level mathematics courses, such as logic, set theory, functions, relations, cardinality, the basic properties of numbers, and rigorous proofs of material from calculus. The format of the class will be as follows. You will read the text and I will go over some of the material in class. Your main job in the class is to prepare proofs of as many of the exercises in the text (plus any other assigned problems) as possible, and to present those proofs either in writing or at the board during class. (Some times false statements are made, and then your job is to demonstrate that they are false, and if possible, to correct them.) All solutions and presentations should be clear; during in-class presentations, both the instructor and other students are encouraged to ask questions at any time. On occasion, you may work in small groups. You will generally need to read the text and work on the problems before we have discussed them in class. You will also need to be in class so that you can make presentations. Office hours: My office hours will be 10:50-11:50 am & 2:20-3:20 pm on Monday & Wednesday. I will generally be available on Tuesdays 1:00-1:50 pm, but you should check because there will be some department meetings during this time. You can also see me by appointment or any time you can find me in my office; do not hesitate to talk to me. All times are subject to change and to cancelation on some days due to other duties. Grades: Your grade will be based on (a) in-class presentations and class participation; (b) written homework; (c) quizzes; (d) midterms, oral exams, and a final exam. The grading scale will be determined based on the class performance (i.e., there will be a curve). The exact portion each of the above activities will contribute to your grade will be decided later. Typically, in-class work and written homework count for at least half the final grade. (Over)

2 Investment of time: A typical student should expect to spend 150 minutes per week in class and at least six hours per week studying and doing homework. The amount of time you need to spend outside of class may vary considerably from this estimate. When taking notes in class and when reading the text or any other material, try to work actively. Anticipate what the next step will be and attempt to come up with your own proofs and your own solutions. Other information: Links to UWM policies relevant to this class can be found at If you have any special requirements or concerns regarding this course, please let me know as soon as possible. Friday, October 24 is the last day to drop the class with a W on your transcript. For other important dates, see staff/instructional_support/registrars_calendar.cfm?term=2149. Homework: Homework will be assigned, collected, and graded regularly. There are many exercises in the text and other problems may be given. It is vital that you work on a wide selection of them, including those that are not assigned to be turned in or presented. It is impossible to really learn mathematics without doing problems! You are encouraged to to come to my office hours to talk about problems and to review draft versions of your homework with me. You are encouraged to discuss homework problems with other students, except that homework that is handed in for a grade must be your work. Please remember that if you don t do it yourself, you won t learn it. Exams: I anticipate that there will be two or three midterm exams during the semester and a comprehensive in-class written final exam. The first midterm exam will be in October. One of the exams may be an oral exam. The final exam will take place on Wednesday, December 17 from 10 am to noon. You cannot take the final early. A make-up exam will not be given without a very good, documented reason that is acceptable to me. If you cannot come to an exam (for that very good reason), please let me know as far in advance as possible: you may call me, me, or leave a message at the Mathematics office, More on the class format and goals: A great deal of emphasis will be placed on developing the ability to write and present organized and rigorously correct proofs. This is a very big job, even for a professional mathematician. You should not be discouraged if it takes several weeks to get the hang of it, or if I or other students ask a lot of questions or make comments or suggestions about your presentations or your written problems. Asking questions and making comments is part of my job as a teacher and your job as a student when you are trying to understand a presentation. All such comments should be constructive and respectful. It is inevitable that we will find serious errors in some presentations, errors that make it unlikely the problem can be solved in the manner outlined. In that case, I may ask someone else to present a solution. At times, I may ask several students to write solutions to the same problem. Please do not take personally either comments or switching to another person for a solution mistakes and even failure are a part of learning mathematics. I hasten to add that one of the goals of the class is to know when a proof (by you or someone else) is correct, so do not claim a problem if you are not confident that you are prepared to present it clearly and correctly.

3 This final exam was given in a previous Math 341 class. It is included to help you get a sense of what the class covers. Keep in mind: (1) the exams are not the most important part of the class and (2) the exams and the specific topics in Fall 2014 may be very different from this exam. MATH 341 Final Exam May 14, 2008 Give precise and complete statements and proofs. You must explain and justify each answer. Unless othewise stated, you can use any material from the book that comes before the fact you are trying to prove. For example, you cannot prove A (B C) = (A B) (A C) by citing Theorem 4.3.2, since this is part of Theorem (4) Neither of the following statements involving the sets A, B makes sense. Explain why not. (i) A B A B (ii) A B = (A B) 2. (7) Let P (x) and Q(x) be open sentences (predicates). Prove that if { x P (x) } { x Q(x) }, then x(p (x) Q(x)). [This is part of Theorem 4.2.3, but you cannot use that theorem in the proof.] 3. (8) Let f : A B, g : B C be functions. Prove that if f, g are onto, then g f is onto. 4. (8) Prove by induction on n that for all positive integers n, n = n. 5. (7) Is the following proof correct, or does it have errors? If there are errors, identify them. Fact: For any sets A, B, if A B B A, then A B = B A. Proof. Assume A B B A. Let a A and b B. Then (a, b) A B B A, so (a, b) = (b, a ) for some a, b. Thus a = b B and b = a A. This shows A B and B A, so A = B. It follows that A B = A A = B B = B A. QED. 6. (6) Let A be the set of people living in Milwaukee and B the set of brands of beer. Is f : A B, given by the rule f(x) = x s favorite brand of beer, a well-defined function? 7. (7) Recall that x y iff there exists an integer n with y = nx. Suppose that a, b, c are integers and a b and a (b 2 c). Prove that a c. [You do not need to use anything but the definition of and basic properties of arithmetic to do this problem.] 8. (7) Define a relation on R by x y if y = nx for some n N. Is an equivalence relation on R? If it is, show that it is. If it is not, show that it isn t.

4 9. (10) Prove that for x, y Z, if xy is odd, then x, y are both odd. In this problem you may not assume facts about the products of odd or even numbers; you need to prove anything you use. (You may assume every integer is either even or odd, but not both.) 10. (8) (a) Find sets A, B with A B = {(1, 2), (1, 3)}. (b) Prove that there do not exist sets A, B with A B = {(1, 1), (2, 2)}. 11. (7) Let f : A B, g : B C be functions. Prove that if g f is onto and g is one-to-one, then f is onto. 12. (8) Prove that 2 is irrational. 13. (12) Are the following logical statements true or not? True means true for all possible P, Q, R, P (x, y). You do not need to justify your answers in this problem. (a) [P (Q R)] [(P Q) R]. (b) [(P Q) P ] [P Q]. (c) [ y xp (x, y)] [ x yp (x, y)]. 14. This problem is a bonus problem, worth 12 points of extra credit. In this problem, we are interested in sets that are subsets of their power sets. Such sets are rare, but they exist. (a) Explain why if A =, then A P(A). (b) Suppose A P(A) and let B = A {A}. Show that B P(B). (c) Explain how to use parts (a) and (b) to generate infinitely many sets A with A P(A). (Write down a few.) (d) Suppose that A = {A}. Show that A P(A). (Whether such a set A exists depends on the axiom system you use for set theory.)

5 MATH 341 Final Exam Solutions May 14, 2008 These are sample solutions. problem. Remember, there may be other ways to solve any given 1. A B is a set, while A B is a statement that is true or false. Thus they cannot be connected by either or =. What was meant here? We can only speculate perhaps A B = A A B? 2. Let a be any [legal] value for x. If P (a) is true, then a { x P (x) }, so a { x Q(x) }, i.e., Q(a) is true. This is true for all possible values a, so x(p (x) Q(x)). 3. Let c C. Since g is onto, there exists b B with g(b) = c. Since f is onto, there exists a A with f(a) = b. Thus g f(a) = g(f(a)) = g(b) = c. This proves g f is onto. 4. Base case: If we let n = 1, the equation becomes = 2 1, which is true since 2 2 both sides equal Induction step: Suppose n = n. Then n n+1 = n n+1 = n n+1 = n+1, which is exactly what we want. 5. There are two problems. The main one is the second sentence of the proof. We may not be able to take a A and b B if either A or B is. In fact, it is possible to have A B if one of them is empty. The second problem is not an error but an omission: the last sentence of the first paragraph should say (a, b) = (b, a ) for some a A, b B. 6. The function is not well-defined, since even in Milwaukee there may be people who do not like any brand of beer (like babies or my wife!) There may also be people who like several brands of beer and don t have a single favorite. Either of these problems disqualifies f from being a function. 7. Since a b and a b 2 c, there exist integers n, m with b = na and b 2 c = ma. Thus b 2 = bna and b 2 + c = ma. Add these equations to get c = bna ma = (bn m)a. Since bn m Z, this shows a c. 8. is reflexive and transitive but it is not symmetric, so it is not an equivalence relation. For example π 2π (take n = 2), but 2π π is false. [You could also use 1 2 but (2 1) as your counterexample.] 9. If x is even, say x = 2n, then xy = 2ny = 2(ny) is even. Likewise, if y is even, then xy is even. Thus xy can only fail to be even if both x, y are odd. You weren t asked to prove this, but if x, y are odd, then xy is. Let x = 2n + 1, y = 2m + 1. Then xy = 2(2nm + n + m) + 1.

6 10. (a) A = {1}, B = {2, 3} is the unique solution. (b) Suppose A B = { (1, 1), (2, 2) }. Then (1, 1) = (a, b) some a A, b B, and so 1 A and 1 B. Using (2, 2) A B, we see in a similar manner that 2 A, 2 B. But this means (1, 2) A B. This contradicts our original assumption. 11. Let b B, so g(b) C. Since g f is onto, there exists a A with g(b) = g f(a) = g(f(a)). Since g is one-to-one, we conclude from this equation that b = f(a). This proves f is onto. 12. Suppose to the contrary that 2 = a b is rational, where we may presume a b is in lowest terms. Squaring this equation and clearing fractions, we see 2b 2 = a 2. Thus a 2 is even (i.e., divisible by 2), so by Corollary or results discussed in class, a is even. Let a = 2n. Then we have 2b 2 = (2n) 2 = 4n 2. Dividing by 2 gives b 2 = 2(2n 2 ), so b 2 is even. As above, this implies b is even. Thus both a, b are even, so a b 10 is not rational. is not in lowest terms. This contradiction shows 13. (a) Not true: if P, Q, R are all F, then P (Q R) is T while (P Q) R is F. (b) This is true. You can check it with a truth table, or algebraically, or note that the LHS of is false precisely when P is T and Q is F, just as P Q is. (c) This is true. The LHS of says there is some particular y 0 such that P (x, y 0 ) is true for every x. Thus for every x, there is a y for which P (x, y) is true, namely y = y 0. This means the RHS is true. 14. (Bonus question) (a) is a subset of any set. (b) By assumption, A P(A). We have A P(A), so {A} P(A). Thus by Theorem 4.3.1b, B = A {A} P(A). Since A B, we have P(A) P(B) (Theorem 4.4.5). Transitivity of (Theorem 4.2.2) now gives B P(B). (c) Starting with (a) and building up as in (b), we get the following sets that are subsets of their power sets:, { }, {, { } }, {, { }, {, { }} }, etc. Note that these sets have 0,1,2,3 elements respectively, and each new set adds one more element. We can go on forever like this, and create infinitely many sets with the desired property. (d) Suppose A = {A}. Then the only element of A is A and we have A A (i.e., A P(A) ). This shows A P(A). As noted, you can t construct such a set A by any straightforward means. Some set theories, such as the standard ZFC, explicitly rule out this possibility, but other axiom systems allow it.