Math 508 Exam 1 - October 12, 2006, Exams of Design and Analysis of Algorithms

Download Math 508 Exam 1 - October 12, 2006 and more Exams Design and Analysis of Algorithms in PDF only on Docsity! Math 508 Exam 1 Jerry L. Kazdan October 12, 2006 12:00 – 1:20 Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5 points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems (60 points, 15 points each). Closed book, no calculators – but you may use one 3′′ × 5′′ card with notes. Part A: Examples (4 problems, 5 points each). Give an example of an infinite set in a metric space (perhaps R) with the specified property. A–1. Bounded with exactly two limit points. A–2. Containing all of its limit points. A–3. Distinct points {xj} , j = 1, 2, . . . with xi 6= xj for i 6= j that is compact. A–4. Closed and bounded but not compact. Part B: Classify sets (20 points) For each of the following sets, circle the listed properties it has: a) {1 + 1 n ∈ R, n = 1, 2, 3, . . .} open closed bounded compact countable b) {1} ∪ {1 + 1 n ∈ R, n = 1, 2, 3, . . .} open closed bounded compact countable c) {(x, y) ∈ R2 : 0 < y ≤ 1} open closed bounded compact countable d) {(x, y) ∈ R2 : x = 0} open closed bounded compact countable e) {(x, y) ∈ R2 : x2 + y2 = 1} open closed bounded compact countable f) {(x, y) ∈ R2 : x2 + y2 ≤ 1} open closed bounded compact countable g) {(x, y) ∈ R2 : y > x2} open closed bounded compact countable h) {(k, n) ∈ R2 : k, n any positive integers} open closed bounded compact countable 1