MATH c UNIVERSITY OF LEEDS Examination for the Module MATH2210 (May/June 2004) INTRODUCTION TO DISCRETE MATHEMATICS. Time allowed : 2 hours

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1 This question paper consists of 5 printed pages, each of which is identified by the reference MATH MATH No calculators allowed c UNIVERSITY OF LEEDS Examination for the Module MATH2210 (May/June 2004) INTRODUCTION TO DISCRETE MATHEMATICS Time allowed : 2 hours Do not answer more than FOUR questions All questions carry equal marks. 1. (a) A bridge hand consists of 13 cards drawn from a pack of 52 cards. i) How many different bridge hands are there? ii) How many different bridge hands are there which do not contain any aces? iii) How many different bridge hands are there which contain no card higher than a nine (with aces counting high)? Such a hand is called a Yarborough. Assuming that a bridge hand is dealt at random, what is the probability (expressed in terms of binomial coefficients) of getting a Yarborough? (b) Let A 1,..., A n be finite sets. State (without proof) the Inclusion-Exclusion Principle giving #(A 1 A n ) in terms of numbers of the form #(A i1 A ik ), with 1 i 1 < < i k n, 1 k n. How many integers are there in the range 1 to 10 6 divisible by none of 2, 3 and 5? (c) Let k be a number between 1 and n, and let D k = the set of permutations of {1,2,..., n} fixing k in its original position. Show that ( n # k=1 D k ) = n! n ( 1) k+1 1 k!. If two packs of cards are dealt one by one simultaneously, what is the probability that there is at least one coincidence of cards? k=1 1 continued...

2 2. (a) Let d n be the number of sequences of length n made up of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, in which two consecutive occurrences of the digit 0 do not appear. Show that d n satisfies the recurrence relation d n+1 = 9d n + 9d n 1. Find the roots α, β of the auxiliary equation for this recurrence relation. Solve the recurrence relation, hence finding a general formula for d n in terms of α, β and n. (b) Let a n be the number of sequences of length n made up from the 26 letters of the alphabet, in which there occur an even number of occurrences of vowels (i.e., of letters taken from the list a, e, i, o, u). Find a non-homogeneous recurrence relation satisfied by a n. Find a solution of this recurrence relation, giving a general formula for a n in terms of n. 3. (a) Use the connectedness algorithm to determine whether the graph given by the following adjacency matrix is connected. v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v 12 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v (b) i) Let G be a graph with ν vertices and ε edges, and with each vertex of degree at least 2. Show that ν ε. ii) Let G be a connected graph with ν vertices and ε edges. Show that ν 1 ε. iii) Deduce that if G is a connected graph with ε = ν 1 edges, then G is a tree. (c) Find a minimal connector G for the following graph and specify the value of M(G ). 2 Question 3 continues...

3 b a edge µ MATH c d e f ab 3 ac 8 ae 10 af 5 bc 7 bd 6 bf 6 cd 4 ce 9 de 8 df 7 ef 9 4. (a) State (without proof) Euler s Formula for a connected planar graph with ν vertices and ε edges which is drawn in the plane with ϕ faces. Prove that if G is a connected planar graph with ν vertices with ν 3, and ε edges, in which all closed paths contain at least 4 edges, then ε 2ν 4. (b) Hence, or otherwise, deduce that the Folkman graph (below) is not planar. 3 Question 4 continues...

4 (c) Prove that a connected planar graph in which all closed paths contain at least 4 edges must have a vertex of degree 3 or lower. (d) Give an example of a connected (non-planar) graph in which all closed paths contain at least 4 edges and in which all vertices have degree at least 4. Say, giving reasons, whether or not there exists such a graph with at most 8 vertices. 5. (a) Consider the URM program: 1. J(1, 3, 10) 2. S(3) 3. J(1,3,9) 4. S(3) 5. J(1,3,9) 6. S(2) 7. S(3) 8. J(1,1,5) 9. T(2,1) i) Draw the flow chart corresponding to this program. ii) Give the full trace table of the URM computation using this program for the single number inputs: 0,1. iii) Find the output of the computation using this program for input 4. iv) Describe the function f : N N computed by this program. (b) Show that the function f(m, n) = m + n is primitive recursive. Assuming that proper subtraction is primitive recursive, show that so is { m n if m > n m n = 0 otherwise { m if m n max(m, n) = n otherwise. (c) Show that every URM computable function f : N N is dominated by a strictly increasing URM computable function g. [You may assume that the set of all URM computable functions is closed under substitution and primitive recursion.] 4 continued...

5 6. The Busy Beaver Function is defined by B(n) = the maximum output, for input 0, of any URM program with at most n instructions. (a) Show that B is strictly increasing. (b) Show that for all n 1, B(n + 5) 2n. (c) Let g be a URM computable function, computed by a URM program P g of length l(p g ) = k 0. Show that for each number n 0 there is a URM program with at most n + k 0 instructions which gives output g(b(n)) from input 0. Deduce that B(n + k 0 ) g(b(n)), for each n 0. (d) Show that if g is known to be strictly increasing, then B(n k 0 ) g(2n), for each n 1, and hence that B dominates g. (e) Deduce that B is not URM computable. [You may assume the result of question 5, part (c).] END 5