Mathematics for Computer Science MATH 1206C [P.T.O] Page 1 of 5

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2 Question 1: (25 Marks) ANSWER ALL QUESTIONS (a) Prove by mathematical induction that for n 1, n r=1 (b) State and prove the Well-Ordering Principle. r 3 = n2 4 (n + 1)2. (c) Using proof by contradiction, prove that 3 2 is irrational. (d) Use the Pigeonhole Principle to show that among any n + 1 numbers one can find 2 numbers so that their difference is divisible by n. [ = 25 marks] Question 2: (25 Marks) (a) Use truth table to establish whether the statement forms a tautology or a contradiction. (b) Show that (p ( p q)) p q. ((q r) ( p q)) q (c) (i) Let f be a function from the set A to the set B. Let V and W be subsets of A. Show that f(v W ) = f(v ) f(w ). (ii) Let P = {1, 2, 3}. Then R = {(1, 1), (1, 2), (1, 3)} is a relation on P. Which of the properties reflexive, symmetric, and transitive does the relation R possess? Justify your answers. (iii) Determine the truth value of the statement x y U, x 2 + y 2 < 12 where U = {1, 2, 3} is the universal set. Page 2 of 5

3 (d) Show that the following recurrence relation a n + 9a n 2 = 6a n 1, a 1 = 3, a 2 = 27, has general solution of the form and give the values of α 1 and α 2. (α 1 + α 2 n)3 n, [ = 25 marks] Question 3: (25 Marks) (a) (i) Let A k denote the set of integers in {1, 2,..., 100} that are divisible by k, for a positive integer k and 100 A k =, k where y denotes the greatest integer less than or equal to y and let A k A l = A lcm(k,l), where lcm(k, l) denotes the least common multiple of k and l. Using the principle of inclusion and exclusion, fnd the number of positive integers less than or equal to 100 that are divisible by 2, 3 and 5. (ii) Hence, find the number of positive integers of not greater than 100, which are not divisible by 2, 3 and 5. (b) A food tray in a university cafeteria consists of 5 kinds of bread, 4 kinds of meat, 4 kinds of cheese and 6 kinds of vegetable. You can make your own sandwich by choosing 1 bread, 1 type of meat, 1 kind of cheese and 3 different kinds of vegetable. How many different sandwiches are possible? (c) The proportion of students who use smartphones on university campuses across the country has increased tremendously over the past few years. It is estimated that approximately 90% of students in cohort BCNS18AFT prefer smartphones to tablets. 15 students are to be selected at random from the same cohort. Using the binomial distribution, find the probability that 13 or more students prefer smartphones to tablets. Page 3 of 5

4 (d) Find the binomial generating function of the sequence a = 1, 36, 216,..., 6 n,... (e) Consider the state machine, M 1 = (B, G, q 0 ) where B = {(x, y) x, y N}, G = {(x, y) (x, y ) x, x, y, y N x + y = x + y }, q 0 = (0, 0). (i) Describe the reachable states for M 1. (ii) State whether or not the predicate P (q = (a, b)) := a > b is a preserved invariant for M 1. Justify your answer. [ = 25 marks] Page 4 of 5

5 Question 4: (25 Marks) (a) Define the following terms (i) Degree of a Vertex (ii) Euler Circuit (iii) Hamilton Circuit (iv) Weighted Graph (v) Simple Path (b) (i) Draw K 5 and K 2,5. (ii) Consider a connected undirected graph G = (v, e) and a weight function f : e R. Describe what is meant by a spanning tree of G. (c) (i) Describe Kruskal s algorithm for finding minimum-length of a spanning tree. (ii) The Central Water Authority wants to lay on a water supply to caravans positioned in different regions of the country specifically at Albion, Bagatelle, Curepipe, Dagotière, Ebene, Moka and Plaine Magnien. The distances in kilometers between each region are shown in Table 1. Table 1: Distances in kilometers. From / To Albion Bagatelle Curepipe Dagotière Ebene Moka Plaine Magnien Albion Bagatelle Curepipe Dagotière Ebene Moka Plaine Magnien Use Kruskal s algorithm to determine how the caravans should be connected so that the total length of pipe required is a minimum. [ = 25 marks] ***END OF QUESTION PAPER*** Mathematics for Computer Science MATH 1206C Page 5 of 5

BSc (Hons) Mathematics. Examinations for / Semester 2

BSc (Hons) Mathematics. Examinations for / Semester 2 BSc (Hons) Mathematics Cohort: BM/13/FT Examinations for 2014-2015 / Semester 2 MODULE: FINITE MATHEMATICS MODULE CODE: Duration: 3 Hours Instructions to Candidates: 1. Answer ALL questions. 2. Questions