CS 250/251 Discrete Structures I and II Section 005 Fall/Winter Professor York

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1 CS 250/251 Discrete Structures I and II Section 005 Fall/Winter Professor York Practice Quiz March 10, 2014 CALCULATORS ALLOWED, SHOW ALL YOUR WORK 1. Construct the power set of the set A = {1,2,3} 2. Write down the prime decomposition of Is (mod 8)? Answer yes or no. Show your work. 4. Find the smallest integer in absolute value which is congruent to 511 modulo For m = 11, list two complete residue systems, one consisting of the smallest nonnegative integers and the other consisting of the integers with the smallest absolute values. 6. Given the set U = {1,2,3,4,5,6,7,8,9}, A = {1,2,5,6}, B = {2,5,7}, C = {1,3,5,7,9} Find: (a) (A C)\B (b) (A B) c (c) (B C)\A Show your work. 7. Show that we can have A B = A C without B = C. 8. Given a universal set U and two sets A and B contained in U, draw two Venn diagrams, one depicting A c and the other depicting A\B. Label your diagrams. 9. Determine whether each of the following is a partition of the set N of positive integers. Explain your answers. (a) [{n: n > 5}, {n: n < 5}] (b)[{n: n 2 > 11}, {n: n 2 < 11}] 10. Find the multiplicative inverse of 4 in the prime field Z 7. Show your work. 11. Determine the validity of the following argument: p q, p, q 12. Given the set S = {1,2,3,4,5,6}. Write down 5 partitions of S in which all subsets of every partition have exactly two elements. Use square brackets to denote the partition and braces to denote the sets in the partition. 1

2 13. Using the prescribed letters for each of the component statements, translate the following compound statements into propositional logic notation: a. A: prices go up B: housing will be plentiful C: housing will be expensive If prices go up, then housing will be plentiful and expensive; but if housing is not expensive, then it will still be plentiful. b. A: going to bed B: going swimming C: changing clothes Either going to bed or going swimming is a sufficient condition for changing clothes; however, changing clothes does not mean going swimming. c. A: it will rain B: it will snow Either it will rain or it will snow but not both 14. Construct the truth table for the following expression and state whether or not it is a tautology. (A B) C A (B C) 15. Consider the following relation R on A = {1,2,3}, R = {(1,2), (2,3), (3,3)} Compute the transitive closure, transitive(r) = R of R. Show your work. 16. By two separate means show that the negation of (p q)is equivalent to p q 17. Determine the contrapositive of each statement: a. If John is a poet, then he is poor. b. Only if Marc studies will he pass the test. 18. Use an example to show that A (B C) (A B) (A C). 19. Determine the probability that one or both numbers exceed 4 in the toss of a two fair dice. Explain your solution (a diagram may help). 20. Write a two-column proof of the following using equivalence and derivation rules. Do not use a truth table. You must have the correct justification for each step of your proof. [(P Q) P] Q 21. Compute the value of 3! (3! + 2!) 22. What is the number of binary relations on the set A = {1,2,3,4}? Explain how you determined your answer. 2

3 23. Consider the relation R = {(1,3), (1,4), (3,2), (3,3), (3,4)}on the set A = {1,2,3,4} Write down the matrix M R of R 24. Let A = {1,2,3,4}, B = {a, b, c}, and C = {x, y, z}. Consider the relations R from A to B and S from B to C as follows: R = {(1, b), (3, a), (3, b), (4, c)} and S = {(a, y), (c, x), (a, z)} Write down R 1 and the composition R S as sets of ordered pairs. 25. In a certain high school 600 students purchased tickets to a dance, 300 purchased tickets to a basketball game, and 173 students purchased tickets to both events. How many students purchased tickets to either of the two events? Show your work. 26. Find the cardinal number ofthe set of all functions from A = {a, b, c, d} into B = {1,2,3,4,5}. Explain your answer. 27. Let h be a function from V = {1,2,3,4} into V. h = {(1,2), (2,3), (3,4), (4,1)}. Is h one-to-one, onto, both, or neither? 28. Let S = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}. Let R be the equivalence relation defined by x y (mod 5), i. e. x y is divisible by 5. Find the partition of S induced by R, i. e. the quotient set S R. Use square brackets to denote the collection of sets in the partition. 29. Prove the following wff is a valid argument (using two-column proof) ( x)p(x) ( x)[p(x) Q(x)]. Use two-column proof. 30. Prove the following wff is a valid argument (using two-column proof) ( x)[h(x) M(x)] H(s) M(s) 31. Show why the converse of the following wff is not valid x A(x) xb(x) x A(x) B(x) 32. Prove the following equivalence by writing an equivalence proof: A B A B False 33. Show that the following wff is a tautology by using equivalences to show that the wff is equivalent to True. Show all steps. Also, show it is a tautology by the truth table method. (A B) A B 3

4 34. List the output order for the nodes in the following binary tree resulting from postorder, preorder, and inorder traversals. Correctly label your answers. 35. Questions on graphs a. How many edges in the complete graph on 10 nodes? b. What is the diameter of the star graph? c. How many nodes does the 5-dimensional binary hypercube contain? d. How many edges does the 5-dimensional binary hypercube contain? 36. What is the difference between a full binary tree and a complete binary tree? Draw a diagram of a complete binary tree which is not a full binary tree of depth Let Σ = {0,1}. Write down the language consisting of all palindromes over Σ of length 4. For purposes of this question, the empty string is not a palindrome. How many strings does this language contain? 38. Given five persons: a. Find the number of ways in which five persons can sit in a row. b. How many ways are there if two of the persons insist on sitting next to one another? c. Solve part (a) assuming they sit around a circular table. d. Solve part (b) assuming they sit around a circular table. 39. There are 12 students in a class. Find the number of ways that 12 students can take 3 tests if 4 students are to take each test. 4

5 40. In a class of 30 students, 10 got A on the first test, 9 got A on the second test, and 15 did not get an A on either test. Find the number of students who got: e. An A on both tests f. An A on the first test but not on the second test g. An A on the second test but not on the first test 41. Suppose 32 students are in art class A and 24 students are in biology class B and suppose 10 students are in both classes. Find the number of students who are: h. In class A or in class B i. Only in class A j. Only in class B 42. Find the minimum number of students needed to guarantee that three of them have last names that begin with the same letter. 43. Find the number of distinct permutations that can be formed from all the letters of the word COMMITTEE. 44. Find n if (a) P(n, 2) = 72; (b) P(n, 4) = 42P(n, 2); (c) 2P(n, 2) + 50 = P(2n, 2). Show your work. 45. Draw a diagram of the first 7 rows of Pascal s triangle. Write down the rule that allows you to compute the values in the next row from the values in the current row. 46. Prove by induction: 1 + a + a 2 + +a n = an+1 1, for a 1 and n 1 a Prove by induction: Prove by induction: n2 n = (n 1)2 n (2n 1)(2n + 1) = n 2n Prove by induction: n + 4 n(n + 1)(n + 2) = n(3n + 7) 2(n + 1)(n + 2) 50. Prove by induction: x 2n y 2n is divisible by x + y 5

6 51. Determine the validity of the following argument: p q, p, q 52. Determine the validity of the following argument: p q, q, p 53. Write the negation of each statement as simply as possible: (a) If she works, she will earn money. (b) He swims if and only if the water is warm. (c) If it snows, then they do not drive the car. 54. Consider the conditional proposition p q. Construct the truth tables for the converse, inverse, and contrapositive of p q. Label each truth table. Which of these propositions is logically equivalent to p q? 55. State and prove the Well Ordering Principle. You can assume the principle of mathematical induction has been proven. 56. Draw the expression tree for the following expression [(2 x 3 y) + 4 z] Prove that 2 is not rational. 58. Determine what conclusion, if any, can be reached from the given hypotheses, then write down wffs describing the hypotheses and conclusion and prove the validity of your argument. Some flowers are purple. All purple flowers are small. 59. Determine what conclusion, if any, can be reached from the given hypotheses, then write down wffs describing the hypotheses and conclusion and prove the validity of your argument. All flowers are red or purple. Pansies are flowers. Pansies are not purple. 60. Determine what conclusion, if any, can be reached from the given hypotheses, then write down wffs describing the hypotheses and conclusion and prove the validity of your argument. Some flowers are red. Some flowers are purple. Pansies are flowers. 6

7 61. Determine what conclusion, if any, can be reached from the given hypotheses, then write down wffs describing the hypotheses and conclusion and prove the validity of your argument. Some flowers are pink and have thorns. All thorny flowers smell bad. Every flower that smells bad is a weed. 62. Prove the following wff is a valid argument. ( x)p(x) ( x)q(x) ( x) P(x) Q(x) 63. Prove by induction: Given an alphabet Σ, Prove x, y ε Σ, reverse(x, y) = reverse(y). reverse(x) 64. Write down the operation tables for +,, of a two element Boolean algebra based on the set A = {a, b} 65. Write down the formal description of the Finite State Machine, M 4, whose state diagram is depicted below. Give an informal description of the language that M 4 accepts. 66. Let Σ = {a, b}. Construct the state diagram for a finite state machine that accepts words w of Σ such that w begins with a and ends with b. 7

8 67. What language over the alphabet Σ = {a, b} is accepted by the FSA in the diagram below? 68. Draw a diagram of the bipartite graph K 2, The Fibonacci numbers are defined by F 0 = 0, F 1 = 1, and F n = F n 1 + F n 2 for n 2 Use induction to prove the following statement F n 1 F n+1 F n 2 = ( 1) n n 70. Provide a visualization of the proof that i=1 i = n(n+1) Does the following graph contain an Euler path? If so, describe one such path. 72. Write down the adjacency matrix and edge list representations of the graph in the previous problem. 73. Let A = {1,2,3}and B = {1,2,3,4}. Let f: A B be an injection. Can you construct a function g: B A such that g f is the identity function i: B B? Explain your reasoning. 8

9 74. In the two-dimensional vector space Z 7 Z 7 over the field Z 7, where Z 7 denotes the finite field comprised of the set of elements Z 7 = {0,1,2,3,4,5,6}, with operations of + 7 and 7 defined as addition modulo 7 and multiplication modulo 7, give the coordinates of the line that goes through the points (1,3) and (3,5). 75. For the following wff give an interpretation in which it is TRUE and an interpretation in which it is FALSE. ( x)( y)(p(x, y) P(y, x)) 76. Which of the following statements are equivalent to the statement, Cats are smarter than dogs? Use the predicates: C(x) = "x is a cat" D(x) = "x is a dog" S(x, y) = "x is smarter than y" Write the predicate calculus statement corresponding to the original statement and its proposed equivalent statement. Explain why the statements are or are not logically equivalent. a. Some cats are smarter than some dogs. b. There is a cat that is smarter than all dogs. c. All cats are smarter than all dogs. d. Only cats are smarter than dogs. e. All cats are smarter than any dog. 77. Write each English language sentence as a wff using the Domain = Whole World and the following predicates: G(x) = x is a game M(x) = x is a movie F(x,y) = x is more fun than y f. Any movie is more fun than any game. g. No game is more fun than every movie. h. Only games are more fun than movies. i. All games are more fun than some movie. 78. Classify the occurrences of each of the variables in the following expression as free or bound. xp(x, y) Q(x) 79. Let p(x) mean that x is a person, let c(s) mean that x is a chocolate bar, and let e(x,y) mean that x eats y. For each of the following wffs, write down an expression that reflects the interpretation of the wff. j. x p(x) y c(y) e(x, y) k. y c(y) x p(x) e(x, y) 9

10 80. Let isfatherof(x,y) be x is the father of y, where the domain is the set of all people now living or who have lived. Find the truth value for each of the following wffs: l. x y isfatherof(x, y) m. y x isfatherof(x, y) n. x y isfatherof(x, y) o. y x isfatherof(x, y) 81. Generate the 4-bit binary reflected gray code vertically. 82. Express the output Y as a Boolean expression in the inputs A and B for the logic circuit depicted below. 83. Determine the diameters of all the complete bipartite graphs. 84. Prove x(a(x) B(x)) x(b(x) C(x)) x A(x) C(x) 85. What is wrong with the following proof of xp(x) xq(x) x P(x) Q(x) 1. xp(x) hyp 2. xq(x) hyp 3. P(a) 1,ei 4. Q(a) 2,ei 5. P(a) Q(a) 3,4,conj 6. x P(x) Q(x) 5,eg 86. Why is Z 6 = {0,1,2,3,4,5} with operations addition mod 6 and multiplication mod 6 not a field? What algebraic structure does it have? 87. Imagine that you own a 32-bit computer that has a DRAM memory consisting of 1GB (gigabytes). 1GB = 2 30 bytes. Every time you load a program into your computer, you can think of it as a string in the language L = {0,1} 238. Is the number of programs that can be represented in your computer finite or infinite? 10

11 88. Given two functions, f(n) = 5 n and g(n) = n!. Which function ultimately grows faster? 89. Explain how attempts to do polynomial interpolation via finite differences may have provide Isaac Newton with insights into the calculus, say especially with respect to the derivative of the exponential function e x -- i.e note that d dx ex = e x. Generate a finite difference table that illustrates this insight for 2 x. 90. Show that the following set of four matrices forms a group, G, with the operation of matrix multiplication, by showing the group operation table (and closure), indicating which element is the identity element, showing that each element has an inverse and showing associativity. If you recognize this group, give a name for it. a = , b = 0, c = 1, d = Why is the knowledge of discrete mathematics and discrete structures essential to the understanding of computer science? 11