M303. Copyright c 2017 The Open University WEB Faculty of Science, Technology, Engineering and Mathematics M303 Further pure mathematics

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1 Faculty of Science, Technology, Engineering and Mathematics M303 Further pure mathematics M303 TMA J Covers Book F Metric spaces 2 Cut-off date 9 May 2018 You will find instructions for completing TMAs in the Assessment area of the M303 website. Please remind yourself of these instructions before beginning work on this TMA. You can submit this TMA either by post, together with a completed TMA form (PT3), to arrive by the cut-off date given above, or electronically, as a single pdf, using the University s online TMA/EMA service. Some of the questions on this TMA are formative, and some are summative. The formative questions should be thought of as teaching material, rather than assessment material. They are designed to develop and extend your understanding of the topics, while the summative ones place more emphasis on assessing your understanding of the topics. Both types of question will be marked by your tutor, but (unlike the summative questions) marks for the formative questions do not count towards your final grade. There is more information about approaching the formative questions on the website. You should submit your solutions to both sets of questions to your tutor by the cut-off date. The marks allocated to each part of a question are indicated in the margin. Although many of the questions on this assignment require numerical answers, the questions invariably carry method marks. In phrasing the questions we have used the words listed below, which should be interpreted as indicated. prove, show, explain, justify determine, find, devise, calculate, compute deduce solve evaluate, give, write down, list hence Clear reasoning and explanation for all steps are called for. An indication of the method used and all working in arriving at an answer should be given. Clear explanation of how one result follows from another is required. Working must be shown. A numerical answer alone is not sufficient. Answer suffices. No explanation need be given. No marks will be awarded for any alternative method. Copyright c 2017 The Open University WEB

2 TMA 06 Cut-off date 9 May 2018 Part A is summative. These questions assess your knowledge of material in the text. You must not discuss these questions in the forums, and your tutor is not allowed to help you directly. Part B is formative. Doing these questions will help you to understand key concepts covered in the text. You are encouraged to discuss these questions in the module forums and with your tutor (who is allowed to help you). We recommend that you submit at least two of these questions to your tutor for marking. Questions 6, 7 and 8 cover material that is more central to the module, while Questions 9 and 10 are designed to expand your understanding beyond the core material. Your tutor will mark all of the answers that you submit. But note that only your marks for Part A count towards your overall continuous assessment score. Some of these questions ask you for a sketch. When sketching, you should follow the conventions used in the module text by using dashed lines to indicate the boundaries of open sets and solid lines to indicate the boundaries of closed sets. The status of individual points can be indicated by using small circles for excluded points and filled discs for included ones. Part A Summative Question 1 10 marks This question tests your understanding of material covered in Chapter 21. Let d be the Euclidean metric on R 2 given by: d ((x 1, y 1 ), (x 2, y 2 )) = (x 1 x 2 ) 2 + (y 1 y 2 ) 2. Consider these four sets in R 2. W = {(x, y) R 2 : x 2 + y 2 < 4} {(x, y) R 2 : x 2 + y 2 > 4 and x 2 + y 2 9} X = {(x, x) R 2 : x R} {(x, x) R 2 : x R} Y = {(q, 1) R 2 : q Q} Z = {(x, y) R 2 : (x + 2) 2 + y 2 = 4} {(x, y) : (x 2) 2 + y 2 < 4} For each set: (a) Sketch, by hand or otherwise, the set, using a separate diagram for each set. (If you are unsure how to represent something on a diagram, or unable to sketch it, then please describe what you mean in words. You are also free to use a combination of sketching and writing.) [4] (b) Write down whether the set is d-connected or d-disconnected. If the set is d-disconnected, then write down a d-disconnection. [6] page 2 of 6

3 Question 2 6 marks This question tests your understanding of material covered in Chapter 22. Let P = {(x, sin x) R 2 : x R} and let d be the Euclidean metric on R 2 given by d ((x 1, y 1 ), (x 2, y 2 )) = (x 1 x 2 ) 2 + (y 1 y 2 ) 2. (a) Sketch P. [1] (b) Write down a d-open cover of P that does have a finite subcover. [2] (c) Write down a d-open cover of P that does not have a finite subcover. [2] (d) Write down a d P -open cover of P. [1] Question 3 5 marks This question tests your understanding of material covered in Chapter 23. Let d be the Euclidean metric on R given by d(x, y) = x y. (a) Let (a n ) be a real sequence. Show that if (a n ) is a d-cauchy sequence then the sequence (a 2 n) is also a d-cauchy sequence. [3] (b) Give an example of a real sequence (b n ) such that the sequence (b 2 n) is a d-cauchy sequence but where the sequence (b n ) is not a d-cauchy sequence. [2] Question 4 13 marks This question tests your understanding of material covered in Chapters Let d be the Euclidean metric on R given by d(x, y) = x y. Let X R be given by { } 1 X = {0} 5 n : n N. (a) Prove that X is totally d X -disconnected, (where d X is the metric induced on X by the Euclidean metric on R). [5] (b) (i) Prove that X B dx (0, r) is a finite set for each r > 0. [2] (c) (i) Hence, or otherwise, show that X is d X -compact. [2] (iii) Let d 0 be the discrete metric on X given by { 0, if x = y d 0 (x, y) = 1, if x y. Is X d 0 -compact? Explain your answer. [1] Is X d X -complete? Is X d 0 -complete? Briefly justify each of your answers. [3] page 3 of 6

4 Question 5 16 marks This question tests your understanding of material covered in Chapter 24. (a) Let and A = {(x, y) R 2 : x 1 and y 3} B = {(x, y) R 2 : x 4 and y 1}. (i) Sketch A and B. Determine d H (A, B), the Hausdorff distance between the sets A and B. [4] (b) Let K 0 and K 1 be as shown below. (1, 1) (0, 1) ( 1 4, 1) (1, 1) (0, 3 4 ) (1, 1 2 ) (0, 1 4 ) (1, 1 4 ) (0, 0) (1, 0) (0, 0) ( 1 4, 0) ( 3 4, 0) (1, 0) K 0 K 1 (i) Write down four similarities S 1, S 2, S 3, S 4 : R 2 R 2 such that K 1 = 4 S i (K 0 ), i=1 and state their similarity ratios. [3] Let K n+1 = 4 i=1 S i(k n ) for n N. Sketch (or describe) the set K 2. [2] (iii) Let K = n=1 Sn (K 0 ), where S : K(R 2 ) K(R 2 ) is given by S(F ) = 4 i=1 S i(f ). Prove that K is the non-empty compact invariant set of the iterated function scheme defined by S 1, S 2, S 3 and S 4. [2] (iv) Hence determine dim(k). [5] page 4 of 6

5 Part B Formative Question 6 7 marks This question revises material from Book D. Let d: N N R be the distance function on N given by d(i, j) = i i + 1 j j + 1. Prove that d is a metric. [7] Question 7 13 marks This question revises material from Book D. (a) Let A = { } 1 R : p N is prime, p and let d be the Euclidean metric on R. (i) Determine whether A is d A -open, d A -closed, both d A -open and d A -closed, or neither d A -open nor d A -closed. [2] Determine whether A is d-open, d-closed, both d-open and d-closed, or neither d-open nor d-closed. [2] (iii) Write down the d-interior and the d-closure of A. [2] (b) For each of the following, determine whether the set is d-open, d-closed, both d-open and d-closed, or neither d-open nor d-closed, as a subset of R 2, where d is the usual Euclidean metric on R 2. (i) X = {(x, y) R 2 : x 1, y 100 and y Z} [2] X = {(x, y) R 2 : x 1 and y Z} [2] (iii) X = {(x, y) R 2 : x 1 and y Q} [3] Question 8 8 marks This question tests your understanding of material covered in Chapter 22. Let A = {f C[0, 1] : f(x) exp(x) for each x [0, 1] and f(x) f(y) x y for x, y [0, 1]}. Show that A is d max -sequentially compact. [8] page 5 of 6

6 Question 9 10 marks This question tests your understanding of material covered in Chapter 23. In Exercise 2.2 of Chapter 15, we introduced the metric space of eventually zero sequences, (l 0, d), where l 0 = {(a n ) : a n R for each n, and (a n ) is eventually zero} and d: l 0 l 0 R is given by d(a, b) = a n b n, where a = (a n ) and b = (b n ). n=1 In this question, we investigate (l 0, d) further. (a) Let p be a prime and let (a k ) be the sequence of points in l 0 given by ( 1 a k = p, 1 p 2, 1 p 3,..., 1 ), 0, 0, 0,... for k N. pk (i) Calculate d(a 10, a 9 ). [1] Find d(a k+1, a k ) for each k N. [2] (iii) Hence show that (a k ) is a Cauchy sequence. [4] (b) Use your results from part (a) to prove that (l 0, d) is not a complete metric space. [3] Question marks This question is designed to help you to revise some topics from earlier books, and takes a second look at the metric on a group introduced in Question 10 of TMA 04. You do not need to have attempted Question 10 of TMA 04 to be able to do this question. Let G be a group, let n N, and suppose that H 0, H 1, H 2,..., H n are subgroups of G such that {e} = H 0 H 1 H 2 H n = G, with H i H i 1 for i = 1, 2,..., n. We define a metric d G : G G R by d G (g, h) = k where gh 1 H k and gh 1 / H i for all i < k. (You may assume that d G is a metric for G.) (a) (i) For g G, describe B dg (g, 1), B dg [g, 1] and S dg (g, 1). For g G, describe B dg (g, 2), B dg [g, 2] and S dg (g, 2). [7] (b) Describe the d G -closed and d G -open sets of G. [2] (c) What conditions, if any, are necessary and sufficient for a group G to be each of the following? (i) d G -connected d G -compact (iii) d G -complete [3] page 6 of 6

MAS331: Metric Spaces Problems on Chapter 1

MAS331: Metric Spaces Problems on Chapter 1 1. In R 3, find d 1 ((3, 1, 4), (2, 7, 1)), d 2 ((3, 1, 4), (2, 7, 1)) and d ((3, 1, 4), (2, 7, 1)). 2. In R 4, show that d 1 ((4, 4, 4, 6), (0, 0, 0, 0)) =