Discrete Mathematics 2 Exam File Spring 2012

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1 Discrete Mathematics 2 Exam File Spring 2012 Exam #1 1.) Suppose f : X Y and A X. a.) Prove or disprove: f -1 (f(a)) A. Prove or disprove: A f -1 (f(a)). 2.) A die is rolled four times. What is the probability that it comes up 3 at least once? 3.) Prove that is irrational. 4.) A box contains eight red apples, six green apples and five plaid apples. a.) You reach into the box and pull out an apple at random. What is the probability the apple is plaid? You reach into the box and pull out an apple at random. You then pull out another apple. What is the probability that both apples are green? c.) You reach into the box and pull out an apple at random. You then pull out another apple. What is the probability that both apples are the same color? 5.) Suppose w, x, y and z are integers. Find the number of ways we can have w + x + y + z = 10 if: a.) w, x, y and z are non-negative. w, x, y and z are positive. c.) w and x are positive, y and z are non-negative. 6.) A class has 12 girls and 10 boys. a.) If a five person committee is chosen at random, how many committees are possible? If a five person committee is chosen at random, how many committees are possible if the committee must have three girls and two boys? 7.) Ignoring years and February 29, what is the probability that, in a group of 6 people, two people share the same birth date? 8.) Three dice are tossed. What is the probability that the sum is greater than 5? 9.) Suppose f: A B and g:b C are one-to-one functions. Prove that g f is one-toone. Exam #2 1.) An inventory consists of a numbered list of 89 items, each marked "available" or "unavailable." There are 50 available items. Show that there are at least two available items in the list exactly nine items apart on the list. (For example, available items at positions 13 and 22 or positions 69 and 78 satisfy the condition.) 2.) The Koch Snowflake is a fractal that comes from repeated iterations of a construction on an equilateral triangle. Start with a triangle with each side length 1 (Figure 1). Then, in the middle of each side, put an equilateral triangle with side length 1/3 (Figure 2). Next, on each edge, insert an equilateral triangle of length 1/9 (Figure 3). Continuing, on each edge insert an equilateral triangle of length 1/27. Continue.... Find a recurrence relation for S n = number of edges in step n. Obviously S 1 = 3.

2 3.) Solve the recurrence relation d n = 4(d n-1 - d n-2 ) subject to the initial conditions d 0 = d 1 = 1. 4.) If possible, draw the following graphs. If it is not possible, explain why not. a.) K 5 K 3,5 c.) a bipartite graph with 3 red vertices and 4 white vertices that is not K 3,4 d.) a simple graph with degree sequence 3, 2, 2, 2, 1, 1, 0 e.) a tree with degree sequence 3, 3, 1, 1, 1, 1, 1, 1 5.) For each of the following graphs, indicate whether it contains an Eulerian path and whether or not it contains an Eulerian cycle? Give reasons for your answers A B C D E F K 6 K 3,4

3 6.) For each of the following graphs, indicate whether it contains a Hamilton path and whether or not it contains a Hamilton cycle? Give reasons for your answers A B C D E F K 6 K 3,4 7.) Redraw the first graph in #6 and label the vertices. Construct the adjacency matrix for the graph. Label the rows and columns of the matrix with the vertex labels. Exam #3 1.) Obviously the following graph is planar. Count the numbers of edges, vertices and faces and verify that Euler's formula holds. (in case it is hard to see, everywhere lines intersect a vertex exists)

4 2.) Using the graph from #1, use a breadth-first approach to finding a spanning tree. 3.) Using the graph from #1, use a depth-first approach to finding a spanning tree. 4.) Draw a graph having six edges and eight vertices or explain why one does not exist. 5.) Draw a graph that is acyclic with four edges and six vertices or explain why one does not exist. 6.) Use Prim's algorithm to find a minimal spanning tree. 7.) Show ALL solutions to the four queens problem. Exam #4 1.) For each pair of graphs, define an isomorphism, if possible, between the two graphs. If it is not possible, give a reason why not. a.)

5 2.) Use a system of equations (as in the recent homework) to show that. 3.) A game begins with 8 stones. Each player may remove one or two stones on each move with the one removing the last stone LOSING the game. Draw a graph that illustrates how this game can proceed and deduce a winning strategy, if possible, for the first player. 4.) Another game involved a queen on a chessboard (8 rows, 8 columns). The first player places the queen anywhere in the topmost row. The two players then alternate moves. A legal move is moving the queen any number of moves down or to the left. The queen may NOT be moved up or to the right. The winner is the player who places the queen on the bottom left square. Draw a graph that illustrates how this game can proceed and deduce a winning strategy, if possible, for the first player. 5.) Prove that the union of two countably infinite sets is countable. 6.) Prove that is irrational. 7.) Consider the following rooted tree. a.) c.) Give the preorder traversal of the tree. Give the inorder traversal of the tree. Give the postorder traversal of the tree. Final Exam 1.) For each of the following numbers, determine, with proof, whether the number is rational or irrational. a.) 12

6 2.) For each pair of graphs, define an isomorphism, if possible, between the two graphs. If it is not possible, give a reason why not. a.) 3.) Suppose w, x, y and z are integers. Find the number of ways we can have w + x + y + z = 14 if: a.) w, x, y and z are non-negative. w, x, y and z are positive. c.) w, x and y are positive and z is non-negative. 4.) A class has 12 girls and 10 boys. a.) If a five person committee is chosen at random, how many committees are possible? If a five person committee is chosen at random, how many committees are possible if the committee must have three girls and two boys? 5.) If possible, draw the following graphs. If it is not possible, explain why not. a.) K 5 K 3,5 c.) a bipartite graph with 3 red vertices and 4 white vertices that is not K 3,4 d.) a simple graph with degree sequence 3, 2, 2, 2, 1, 1, 0 e.) a tree with degree sequence 3, 3, 1, 1, 1, 1, 1, 1 6.) Suppose f: A B and g:b C are one-to-one functions. Prove that g f is one-toone. 7.) A game involves a queen on a chessboard (8 rows, 8 columns). The first player places the queen anywhere in the topmost row. The two players then alternate moves. A legal move is moving the queen any number of moves down or to the

7 left. The queen may NOT be moved up or to the right. The loser is the player who places the queen on the bottom left square. Deduce a winning strategy, if possible, for the first player. 8.) What is the probability that in a group of six people, two will have birthdays in the same month? (Assume all months are equally likely) 9.) In your drawer you have 11 identical plaid socks, six identical blue socks, two identical green socks and five identical red socks. You grab socks out of the drawer without looking. a.) How many socks must be pulled from the drawer to be certain of having a matching pair of socks? How many socks must be pulled from the drawer to be certain of having four matching socks (my dog will be wearing them)? c.) How many socks must be pulled from the drawer to be certain of having a matching pair of green or plaid socks? 10.) a.) Prove that for every connected graph, G, if G has no cycles, then for every pair of vertices a and b in G, there is only one path from a to b in G. Prove that, in a simple, connected graph, two vertices must have the same degree.

2.) From the set {A, B, C, D, E, F, G, H}, produce all of the four character combinations. Be sure that they are in lexicographic order.

2.) From the set {A, B, C, D, E, F, G, H}, produce all of the four character combinations. Be sure that they are in lexicographic order. Discrete Mathematics 2 - Test File - Spring 2013 Exam #1 1.) RSA - Suppose we choose p = 5 and q = 11. You're going to be sending the coded message M = 23. a.) Choose a value for e, satisfying the requirements