The candidates are advised that they must always show their working, otherwise they will not be awarded full marks for their answers.

Transcription

1 MID SWEDEN UNIVERSITY TFM Examinations 2006 MAAB16 Discrete Mathematics B Duration: 5 hours Date: 7 June 2006 There are EIGHT questions on this paper and you should answer as many as you can in the time allowed. Each question carries 3 marks and the maximum mark for the paper is 24 marks. The candidates are advised that they must always show their working, otherwise they will not be awarded full marks for their answers. The candidates are further advised to start each of the eight questions on a new page and to clearly label their answers. This is a closed book examination. No books, notes or mobile telephones are allowed in the examination room. Electronic calculators which cannot display text and formulas may be used. Write the make and model of your calculator on the front of your script. GOOD LUCK!! 1

2 Part A: Theory Question 1 (a) Define Euler s Φ-function. (b) Showing your working, compute Φ(18). (c) State a formula which computes Φ(n) from the unique prime factorisation of n. (d) State (without proof) the Sieve Principle for n sets and sketch a proof for the formula from (c) which uses the Sieve Principle. Question 2 (a) What is a binomial coefficient? (b) Give a combinatorial argument to show that ( ) ( ) n n = k n k for all positive integers n and k where 0 k n. (c) Give a combinatorial argument to show that ( ) ( ) ( ) n n 1 n 1 = + k k k 1 for all positive integers n and k where 1 k < n. (d) What is Pascal s Triangle? 2

3 Question 3 Let n 2 be an integer. (a) (i) Give the definition of a Latin Square of order n. (ii) Show that the multiplication table for the non-zero elements of Z n is a Latin Square of order n 1 if and only if n is a prime. (b) (i) Give the formal condition for two Latin Squares A = [a ij ] and B = [b ij ] where i, j = 1, 2,..., n on symbols S = {1, 2,... n} to be orthogonal. Also explain the concept of orthogonality of Latin Squares in your own words. (ii) Prove that there does not exist a pair of orthogonal Latin Squares of order 2. (iii) Construct four mutually orthogonal Latin Squares of order 5, taking care to explain your method of construction. Question 4 Let G = (V, E) be a graph with edge set E and vertex set V and let d denote the maximum degree of G. (a) (i) What is an edge colouring of G? (ii) Prove that any edge colouring of G must use at least d colours. (b) Suppose now that G is bipartite such that its vertex set V = X Y where X and Y are disjoint. (i) Prove by induction on the number of edges of G that d colours are sufficient to edge colour G. Illustrate the induction step in your proof by a good, non-trivial example. (ii) Explain how to edge colour the complete bipartite graph K n,n using just n colours. Illustrate your explanation by edge colouring K 5,5. (c) Justifying your answer, say whether the result from (b)(i) also holds when G is not bipartite. 3

4 Part B: Problems Question 5 Answer the following questions, justifying your answer in each case. (a) Does 61 divide ? (b) Find r Z such that 0 r 4096 and r (mod 4097). Hence prove that 4097 is a composite integer without factorising it. (c) How many invertible elements are there in Z 121? (d) Suppose we let a computer pick elements from N 200 at random without repetition. How many elements must we let it pick in order to make sure that it has picked a pair of elements whose sum is precisely 201? Question 6 Showing all your working, compute the following numbers. (a) The Stirling number S(7, 3), i.e. the number of partitions of a 7-set which have 3 parts. (b) The number of surjective functions from Z 3 to Z 7. (c) The number of surjective functions from Z 7 to Z 3. (d) The number p(7), i.e. the number of partitions of the positive integer 7. 4

5 Question 7 Let α and β be two permutations in S 8 given by the tables x α(x) x β(x) (a) Write α and β in disjoint cycle notation. (b) Find α 1 β. (c) Proving your answer, say whether α is an even or an odd permutation. (d) Proving your answer, say whether α 3 β 7 is an even or an odd permutation. (e) Showing your working, find σ S 8 such that ασα = σ. Question 8 (a) Let G be a bipartite graph and let M be a matching in G. Say what it means for a path in G to be an M-alternating path which can be used to augment the matching M. (b) Consider the bipartite graph G with the following adjacency list. a b c d e f g h (i) Use augmenting paths to find a maximum matching in G starting from the matching M 0 = {b5, c2, d8, e3, h1}. (ii) State Hall s condition for the existence of a complete matching in G and use it to prove that G does not have a complete matching. 5