For each of the following, state whether the statement is true or false and give a...

 

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For each of the following, state whether the statement is true or false and give a brief justification. You may assume that inputs to the functions belong to the domain of the function (i.e logn implies that we only consider n > 0.) unless otherwise stated. Note that we write lgm n = (Ign)m. (a) ex 1+x for all x R. (b) 2lgn = n. (c) If natural numbers n < m, then n! divides m!. (d) lg n = (lge) (Inn). 2. [10 marks] Use the definition of O, N and E, to prove the following; do not use limits. In your solution, give sufficient details to carefully argue why the constants that you have chosen guarantee that the inequality holds in the definition of order notation. (a) 3n+1/1gn O(n). (b) n 5n N(n). 3. [10 marks] Prove or disprove each of the following. Either give a proof using the definitions of order notation, or give a counter example. 1 Human Sexuality: A X D2L assignment1 - CSCI3 x D2L Course syllabus - CS X | colouring night - G... A -O Attributes and Dimer dal.brightspace.com/d21/le/content/306781/viewContent/4132369/View 3110 coursehero 3110 midterm prep midterm 2 TU Ch2.pdf 2016 midterm.pdf 2 of 2 G Security and Usabilit Privacy error Automatic Zoom ACLs Assignment Assignment 6 Learn Java | Onlin... (a) If f(n) = O(g(n)) and f2(n) = O(g(n)), then f1(n) + f2(n) = O(g(n) + 9(n)). (b) If f(n) = (g(n)) and f2(n) = O(g(n)), then f(n) f2(n) 0(1). 4. [10 marks] Show that for all natural numbers m, ln n = o(n). 5. [10 marks] During lecture, we saw the stable marriage problem. Suppose that the input is further restricted so that the preference list for all men are the same, and the preference list for all women are the same. Prove that there exists a unique stable marriage. To reiterate the restricted problem is the following: Input: n men: m, ...mn, n women: W,... Wn, and a preference list for each person. For all men, their preference list of the women are the same. For all women, the preference lists of the men are the same. Output: A set of n marriages {(W, Mj), ..., (Win, min)} such that everyone is married, and the marriages are stable. 6. [10 marks] Bubblesort is a sorting algorithm that can be described as: repeatedly make passes through the list, comparing adjacent elements, and swap them if they are out of order. Stop when a pass does not swap any elements. Dashboard (a) Give pseudocode for bubblesort. (b) Analyze the best case and worst case runtimes of bubblesort using O-notation. X | + B All Bookmarks For each of the following, state whether the statement is true or false and give a brief justification. You may assume that inputs to the functions belong to the domain of the function (i.e logn implies that we only consider n > 0.) unless otherwise stated. Note that we write lgm n = (Ign)m. (a) ex 1+x for all x R. (b) 2lgn = n. (c) If natural numbers n < m, then n! divides m!. (d) lg n = (lge) (Inn). 2. [10 marks] Use the definition of O, N and E, to prove the following; do not use limits. In your solution, give sufficient details to carefully argue why the constants that you have chosen guarantee that the inequality holds in the definition of order notation. (a) 3n+1/1gn O(n). (b) n 5n N(n). 3. [10 marks] Prove or disprove each of the following. Either give a proof using the definitions of order notation, or give a counter example. 1 Human Sexuality: A X D2L assignment1 - CSCI3 x D2L Course syllabus - CS X | colouring night - G... A -O Attributes and Dimer dal.brightspace.com/d21/le/content/306781/viewContent/4132369/View 3110 coursehero 3110 midterm prep midterm 2 TU Ch2.pdf 2016 midterm.pdf 2 of 2 G Security and Usabilit Privacy error Automatic Zoom ACLs Assignment Assignment 6 Learn Java | Onlin... (a) If f(n) = O(g(n)) and f2(n) = O(g(n)), then f1(n) + f2(n) = O(g(n) + 9(n)). (b) If f(n) = (g(n)) and f2(n) = O(g(n)), then f(n) f2(n) 0(1). 4. [10 marks] Show that for all natural numbers m, ln n = o(n). 5. [10 marks] During lecture, we saw the stable marriage problem. Suppose that the input is further restricted so that the preference list for all men are the same, and the preference list for all women are the same. Prove that there exists a unique stable marriage. To reiterate the restricted problem is the following: Input: n men: m, ...mn, n women: W,... Wn, and a preference list for each person. For all men, their preference list of the women are the same. For all women, the preference lists of the men are the same. Output: A set of n marriages {(W, Mj), ..., (Win, min)} such that everyone is married, and the marriages are stable. 6. [10 marks] Bubblesort is a sorting algorithm that can be described as: repeatedly make passes through the list, comparing adjacent elements, and swap them if they are out of order. Stop when a pass does not swap any elements. Dashboard (a) Give pseudocode for bubblesort. (b) Analyze the best case and worst case runtimes of bubblesort using O-notation. X | + B All Bookmarks