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.

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Justin Fisher
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1 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 of RSA Encryption, from the set {7, 8}. b.) Calculate the value, C, that you will transmit to the recipient. 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. 3.) There are 4237 Jovians and 3985 Martians in the park. It is impossible to tell the Jovians apart and it is impossible to tell the Martians apart. Four of the Jovians and three of the Martians are sitting around a circular table. In how many ways can the seven entities be seated. 4.) In a box there are eight red marbles, four blue marbles and seven plaid marbles. You take THREE marbles out of the box. a.) If each marble is returned to the box before the rest are pulled, what is the probability that all three marbles are blue? b.) If each marble is returned to the box before the rest are pulled, what is the probability that exactly two of the marbles are blue? c.) If no marbles are returned to the box, what is the probability that all three marbles are plaid? d.) If no marbles are returned to the box, what is the probability that all three are different? 5.) You roll two standard six-sided dice and MULTIPLY the numbers. a.) What is the probability that the product is less than 10? b.) What is the probability that the product is less than or equal to 30? c.) What is the probability that the product is prime? d.) What is the probability that the product is seven? 6.) Suppose a 1 = 4 and a n = 3a n-1 - n, for n > 1. Find the following. a.) a 1 b.) a 2 c.) a 3 d.) a 4 7.) Suppose a 1 = 4 and a n = 3a n-1, for n > 1. Find the following. a.) a 2 b.) a 3 c.) a n (closed, non-recursive form) d.) a 13 (simplified) Exam #2 1.) 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. [hint: Suppose not.]

$2.) The Sierpinski Gasket is a fractal that comes from repeatedly removing portions of an equilateral triangle. Start with a triangle with each side length 1 (Figure 1).$

2 2.) The Sierpinski Gasket is a fractal that comes from repeatedly removing portions of an equilateral triangle. Start with a triangle with each side length 1 (Figure 1). Then, in the middle, remove an equilateral triangle with vertices at the midpoints of the sides of the original triangle (Figure 2). Next, in each remaining triangular portion, remove an equilateral triangle with vertices at the midpoints of the sides of the remaining triangles (Figure 3). Continue.... a.) Find a recurrence relation for S n = number of shaded triangles in step n. Obviously S 1 = 1. b.) Find a closed form expression for S n = number of shaded triangles in step n. 3.) Solve the recurrence relation d n = 5d n-1-4d n-2 subject to the initial conditions d 0 = 2, d 1 = 1. 4.) If possible, draw the following graphs. If it is not possible, explain why not. a.) K 4 b.) 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 simple graph with degree sequence 5, 3, 2, 2. 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

3 D E F K 5 K 2,5 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 5 K 2,5 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. 8.) Draw a graph having six edges and eight vertices or explain why one does not exist.

4 9.) 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.) b.) Exam #3 1.) Use Kuratowski's Theorem to determine if the following graph is planar.

5 2.) Do the following: a.) Define a tree b.) (fill in the blank) If a tree has n vertices then it has edges. c.) (circle answer) True False - A tree contains an even number of cycles. d.) (circle answer) True False - A tree can have the following degree sequence. 3, 3, 2, 2, 1, 1, 1, 1. e.) If the answer to part d.) is "true," draw such a tree. If it is "false," draw your favorite superhero. 3.) Consider the following tree. Consider vertex "a" to be the root. a.) b.) Construct a spanning tree using the breadth-first algorithm. Construct a spanning tree using the depth-first algorithm. 4.) Using Prim's Algorithm, find a minimal spanning tree for the following tree. For #5-7, consider the following tree.

6 5.) Give the order of vertices for a preorder traversal of the above tree. 6.) Give the order of vertices for a inorder traversal of the above tree. 7.) Give the order of vertices for a postorder traversal of the above tree. 8.) Use Kuratowski's Theorem to explain why every tree is planar. 9.) Determine if the following pair of trees is isomorphic. If they are not, explain why not. If they are, label the vertices of the tree on the left using the letters A, B, C, D, E, F, G. Then label the vertices of the tree on the left using the letters a, b, c, d, e, f, g so that the isomorphism is f(a) = a, f(b) = b, etc. 10.) Determine if the following pair of trees is isomorphic. If they are not, explain why not. If they are, label the vertices of the tree on the left using the letters A, B, C, D, E, F, G, H. Then label the vertices of the tree on the right using the letters a, b, c, d, e, f, g, h so that the isomorphism is f(a) = a, f(b) = b, etc.

7 Exam #4 1.) Use the division method to approximate to 3 places to the right of the decimal. 2.) Use the Babylonian method to approximate to 3 places to the right of the decimal. 3.) Convert each number to the indicated base. You may use your calculator to help with arithmetic but ALL necessary work must be shown. a.) 6543 to base 5 b.) to base 10 c.) to base 4 (do enough places to the right of the decimal to be sure of the repeating pattern) 4.) Find the last two digits each of the following. You may use your calculator to help with the arithmetic in finding the pattern but you must show all work. a.) b.) ) Find the coefficient of a 4 b 3 c 2 d in the expansion of (a + b + c + d) ) Evaluate the following fraction (pay close attention to the numbers). Once you get to the polynomial that will solve the problem, give an exact answer for the answer and a decimal approximation using your calculator.

8 7.) Let n = maximum{7, number of letters in your last name}. Construct a primitive Pythagorean Triple using n and any m that you choose. 8.) Evaluate. Once you get to the polynomial Final Exam that will solve the problem, give an exact answer for the answer and a decimal approximation using your calculator. 1.) Prove is irrational. 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. 3.) Consider the number 354,638,838,122,712. WITHOUT USING YOUR CALCULATOR, determine if it is divisible by the following. Give reasons for your answers. a.) 2 b.) 3 c.) 4 d.) 5 e.) 6 f.) 7 g.) 8 h.) 9 i.) 10 j.) 11 k.) 12 l.) 13 4.) In a box there are eight red marbles, four blue marbles and seven plaid marbles. You take THREE marbles out of the box. a.) If each marble is returned to the box before the rest are pulled, what is the probability that all three marbles are blue? b.) If each marble is returned to the box before the rest are pulled, what is the probability that exactly two of the marbles are blue? c.) If no marbles are returned to the box, what is the probability that all three d.) marbles are plaid? If no marbles are returned to the box, what is the probability that all three are different? 5.) You roll two standard six-sided dice and MULTIPLY the numbers. a.) What is the probability that the product is less than 10? b.) What is the probability that the product is less than or equal to 30? c.) What is the probability that the product is prime? d.) What is the probability that the product is seven? 6.) Suppose a 1 = 4 and a n = 3a n-1 - n, for n > 1. Find the following. a.) a 1 b.) a 2 c.) a 3 d.) a 4 7.) Suppose a 1 = 4 and a n = 3a n-1, for n > 1. Find the following. a.) a 2 b.) a 3 c.) a n (closed, non-recursive form) d.) a 13 (simplified) 8.) If possible, draw the following graphs. If it is not possible, explain why not. a.) K 4 b.) 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 simple graph with degree sequence 5, 3, 2, 2. 1, 1

9 9.) Using Prim's Algorithm, find a minimal spanning tree for the following tree. 10.) Convert each number to the indicated base. You may use your calculator to help with arithmetic but ALL necessary work must be shown. a.) 6543 to base 5 b.) to base 10 c.) to base 4 (do enough places to the right of the decimal to be sure of the repeating pattern) 11.) A two-player game begins with 8 stones. Each player may remove one or two stones on each move with the one removing the last stone WINNING the game. Draw a graph that illustrates how this game can proceed and deduce a winning strategy, if possible, for the first player. 12.) 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. b.) w, x, y and z are positive. c.) w and x are positive, y and z are non-negative. 13.) 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.

10 14.) Solve the recurrence relation d n = 4(d n-1 - d n-2 ) subject to the initial conditions d 0 = d 1 = ) Use a system of equations (as in the recent homework) to show that. 16.) 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

Discrete Mathematics 2 Exam File Spring 2012

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