UNIVERSITETET I BERGEN Det matematisk-naturvitenskapelige fakultet. Obligatorisk underveisvurdering 2 i MNF130, vår 2010

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1 UNIVERSITETET I BERGEN Det matematisk-naturvitenskapelige fakultet Obligatorisk underveisvurdering 2 i MNF130, vår 2010 Innleveringsfrist: Fredag 30. april, kl. 14, i skranken på Realfagbygget. The exercises must be solved and handed in individually by each student. Collaboration between students is not allowed. The result counts 10% of the final grade. However, if the result is better than the score at the final exam, the result will count 20%. Aids permitted: Textbook and calculator. All answers must be justified. You should include enough calculations and intermediate steps to make your methods transparent. The justifications must however be short and concise. Unnecessarily long explanations will be rewarded with fewer points. The problem set consists of 5 pages. PROBLEM 1 (20 points) Do one of the following two problems: (I) Let a 1 = 2, a 2 = 9 and a n = 2a n 1 + 3a n 2 for n 3. Show that a n 3 n for all positive integers n. (II) In a round-robin pool tournament there are n participants. (Everyone plays exactly one game against each other participant and a pool game cannot end with a draw.) Show that no matter how the games end, it will be possible after the tournament to make a list of all players such that each player has beaten the next player in the list in the match they played against each other. PROBLEM 2 ( points) (a) How many students must be in a class to guarantee that at least five were born on the same day of the week? (b) How many different license plates can be made if each license plate consists of three letters followed by three digits or four letters followed by two digits? (The letter can be taken from the English alphabet, which consists of 26 letters.) (c) You are at a family dinner, where there are nine other members of your family. You want to arrange five of the other nine persons in a row for a picture. In how many different ways can this be done, if your parents are to be in the picture, standing next to each other? 1

2 2 PROBLEM 3 ( points) Consider the following relations on the set of positive integers: R 1 = {(x, y) gcd(x, y) = 1} R 2 = {(x, y) x and y have the same prime divisors} (a) Check whether the relations are reflexive, symmetric, antisymmetric, or transitive. (b) Why is R 2 an equivalence relation? What is the equivalence class of 1? Write the equivalence class of a positive integer n 2 in the simplest way possible, using the Fundamental Theorem of Arithmetics. (c) What is the intersection of the two relations R 1 and R 2? PROBLEM 4 (10 points) Suppose that R and S are symmetric relations on a set A. Prove that R S is also symmetric. PROBLEM 5 ( points) Consider the poset S with the following Hasse diagram: (a) Why is S not totally ordered? Find a totally ordered subset of S consisting of 5 elements. (b) Find the following elements, or show that they do not exist: (i) all maximal elements of the poset; (ii) all minimal elements of the poset; (iii) the least element of the poset; (iv) the greatest element of the poset; (v) the greatest lower bound of the set {a, b, c}; (vi) the least upper bound of the set {a, b, c}. (c) Use a topological sort to find a total ordering of the elements that is compatible with the partial ordering. PROBLEM 6 ( points) (a) Does a simple graph that has five vertices each of degree three exist? If so, draw such a graph. If not, explain why it does not exist.

3 3 (b) Determine whether the following graph is bipartite or not: (c) The picture shows the floor plan of an office. Use ideas from graph theory to prove that it is impossible to make a walk that passes through each doorway exactly once, starting and ending at A. PROBLEM 7 (10+10 points) Consider the following program segment: i := 1 total:= 1 while i < n begin i := i + 1 total:=total+i end (a) Let p be the proposition total= i(i+1) 2 and i n. Show that p is a loop invariant. (b) Use p to prove that the program segment terminates with the output n(n+1) 2, where n is a positive integer. PROBLEM 8 ( points) (a) What is the language generated by the grammar with vocabulary {S, A, 0, 1}, where S is the start symbol and {0, 1} is the set of terminal elements, and productions S SA, S 0, A 1A and A 1? (b) Find a grammar generating the set {0 2n 1 n n Z, n 0} (with the usual convention that = λ, the empty string).

4 4 (c) What is the output string generated by the finite-state machine described by the following diagram when the input string is 11101? PROBLEM 9 (10+10 points) (a) Here is an incorrect solution to a problem. Find and explain what is not correct and give the correct solution. Problem: Find the number of ways to get two pairs of different ranks (such as two jacks and two fives) in a 4-card hand from an ordinary deck of 52 cards. ( Solution: There are 13 ways to get a rank (such as kings ) for the first pair and 4 ) 2 ways to get a pair of that rank. Similarly, there are 12 ways to get a rank (such as sevens ) for the second pair and ( 4 2) ways to get a pair of that rank. Therefore, there are ( ) ( ) ways to get two pairs. (b) What is the probability that a 5-card hand from an ordinary deck of 52 cards contains two pairs of different ranks? (You will need the correct answer from (a). If you cannot come up with an answer to (a), just call the answer x and give the answer to (b) in terms of x.) PROBLEM 10 ( points) In this problem n is a positive integer. A soccer player has n numbered footballs that he shoots against a goal. He notes with which balls he scores a goal and with which balls he does not. (a) How many different outcomes of the experiment are there? How many outcomes are there with precisely k goals, when 0 k n? (b) A goalkeeper tries to save the shots. He notes which shots go outside the goal, which shots go inside the goal and which shots he saves. How many outcomes of the experiment are there now? (c) Let S be a set of n elements and consider the power set P (S) of S. Show that the set {(A, B) P (S) P (S) A B} contains 3 n elements.

5 5 (d) Give a combinatorial proof of the formula n ( ) n 2 k = 3 n k k=0 (which is Corollary 3 in 5.4, where it is proved using the Binomial Theorem). PROBLEM 11 ( points) In this problem we will develop a proof of the following theorem of Fermat: If p is a prime number, then for any integer n we have n p n (mod p). (a) Let p be a prime number. Prove first the following result using the fundamental theorem of arithmetics: If a and b are integers such that pa/b is an integer and p does not divide b, then pa/b is divisible by p. Then use this result to prove that p divides ( p k) for any integer k such that 0 < k < p. (b) Use the result from (a) and the binomial theorem to prove that, for any integers x and y we have (x + y) p x p + y p (mod p). (c) Use mathematical induction and the result from (b) to prove that n p n (mod p) for all positive integers n. (Hint: n = (n 1) + 1.) (d) Explain why the formula you obtained in (c) is also valid if n is zero or a negative integer. (This finishes the proof of the theorem of Fermat.) PROBLEM 12 (10 points) On a remote island lives a happy tribe. The island is so small that everybody knows everybody. The inhabitants are extremely intelligent and rational, except for the following: It is considered to be a big shame to have blue eyes. If one inhabitant discovers having blue eyes, he/she will flee from the island at midnight and never come back. Luckily there are no mirrors on the island, nobody knows genetics and everybody is too polite to talk about anybody s colour of eyes. One day a missionary comes to the island. As he is about to leave he tells everybody that he finds it strange that there are two colours of eyes in such a small community, blue and brown. On the 43rd night after this happens, many inhabitants flee from the island. How many persons were blue-eyed on the island? (Hint: use induction on the number of blue-eyed inhabitants.) LYKKE TIL! Andreas Leopold Knutsen

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