CS 2102 – Practice problems with sets
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Assume the following definitions:

notation meaning
\mathbb{Z} The integers
\mathbb{Z}^{+} The positive integers; i.e., \big\{ x \; \big| \; x \in \mathbb{Z} \land x > 0 \big\}
\mathbb{N} The natural numbers; i.e., \big\{ x \; \big| \; x \in \mathbb{Z} \land x \geq 0 \big\}
\mathbb{Z}^{-} The negative integers; i.e., \big\{ x \; \big| \; x \in \mathbb{Z} \land x < 0 \big\}
\mathbb{R} The real numbers
\mathbb{Q} The rational numbers; i.e., \Big\{ \frac{x}{y} \; \Big| \; x \in \mathbb{Z} \land y \in \mathbb{Z}^{+} \Big\}
\pi The ratio of the circumference of a circle to its diameter; 3.1415926535…

Assume that \mathbb Q^{+}, \mathbb Q^{-}, \mathbb R^{+}, and \mathbb R^{-} are defined similarly to \mathbb Z^{+} and \mathbb Z^{-}.

1 Membership

1.1 Simple membership

Each of the following is either true or false; which one?

  • 3 \in \mathbb Z1

  • 3.5 \in \mathbb Z2

  • \pi \in \mathbb Z3

  • 3 \in \mathbb Q4

  • 3.5 \in \mathbb Q5

  • \pi \in \mathbb Q6

  • 3 \in \mathbb R7

  • 3.5 \in \mathbb R8

  • \pi \in \mathbb R9

  • 3 \in \{\{1\}, \{2, 3\}, \{4, 5, 6\}\}10

  • \{3\} \in \{\{1\}, \{2, 3\}, \{4, 5, 6\}\}11

  • \{2, 3\} \in \{\{1\}, \{2, 3\}, \{4, 5, 6\}\}12

  • \{2, 3\} \in \mathcal{P}\big(\{2, 3\}\big)13

  • |\{2, 3\}| \in \{2, 3\}14

  • |\{2, 3\}| \in \mathcal{P}\big(\{2, 3\}\big)15

  • \infty \in \mathbb R16

1.2 Closed sets

A set is said to be closed over an operation if applying that operation to members of the set always results in another member of that set.

  • Which (if any, or all) of the following operators is \mathbb Z closed over?
    • 17
    • 18
    • 19
    • 20
    • 21
    • 22
  • Which (if any, or all) of the following operators is \mathbb N closed over?
    • 23
    • 24
    • 25
    • 26
    • 27
    • 28
  • Which (if any, or all) of the following operators is \mathbb R^{-} closed over?
    • 29
    • 30
    • 31
    • 32
    • 33
    • 34
  • Which (if any, or all) of the following operators is \mathbb Q closed over?
    • 35
    • 36
    • 37
    • 38
    • 39
    • 40
  • Which (if any, or all) of the following operators is \mathbb Q \setminus \{0\} closed over?
    • 41
    • 42
    • 43
    • 44
    • 45
    • 46
  • Which (if any, or all) of the following operators is \mathbb R closed over?
    • 47
    • 48
    • 49
    • 50
    • 51
    • 52

2 Comparison

For each of the following, fill in the blank with the first element of the following list that applies:

  • = if the two are identical; otherwise
  • \subset or \supset if those are true; otherwise
  • \subseteq or \supseteq if those are true; otherwise
  • disjoint if the intersection of the two is \emptyset; otherwise
  • \neq
Set 1   Set 2
\mathbb R 53 \mathbb Q
\mathbb N 54 \mathbb Z^{+}
even numbers 55 odd numbers
prime numbers 56 odd numbers
\{1, 3, 5\} 57 \{\{1\}, \{3\}, \{5\}\}
\{1, 3, 5\} 58 \{5, 3, 1\}
\{1, 3, 5\} 59 \{5, 3\}
\mathbb R \setminus \mathbb Z 60 \mathbb R \setminus \mathbb Q
\mathbb Q \setminus \mathbb Z 61 \{1, 2, 4\}
\emptyset 62 \mathcal{P}(\emptyset)
\{1\} 63 \mathcal{P}(\{1\})

3 Listing members and cardinality

For each of the following, list the members of the set:

  • \mathcal P \big(\mathcal P(\emptyset)\big)64
  • \mathcal P \Big(\mathcal P \big(\mathcal P(\emptyset)\big)\Big)65
  • Assume that A = \{25,0,1\}; A \cup \mathcal P(A)66
  • Assume that A is the set of all 2-digit numbers; |\mathcal{P}(A)|67
  • Assume that A is the set of all 2-digit numbers; |\mathcal{P}(A) \cap A|68
  • Assume that A is the set of all 2-digit numbers; |\mathcal{P}(A) \cup A|69

4 Set-builder notation

Assume A = \{1,2,3\} and B = \{2,3,5\}. Write out each of the following in full.

Note that \land means and (like and in Python or && in Java); \lor means or (like or in Python or || in Java); and \lnot means not (like not in Python or ! in Java).

  • \big\{ x \;\big|\; x \in A \big\} 70
  • \big\{ x \;\big|\; x+1 \in A \big\} 71
  • \big\{ x \;\big|\; x \in A \land x \in B \big\} 72
  • \big\{ x \;\big|\; x \in A \lor x \in B \big\} 73
  • \big\{ x \;\big|\; x \in A \land x \notin B \big\} 74
  • \big\{ x+1 \;\big|\; x \in A \big\} 75
  • \big\{ x+y \;\big|\; x \in A \land y \in B \big\} 76
  • \big\{ \{x\} \;\big|\; x \in A \big\} 77
  • \big\{ \{x,y\} \;\big|\; x \in A \land y \in B \land x \ne y \big\} 78
  • \big\{ \{x,y\} \;\big|\; x \in A \land y \in B \big\} 79
  • \big\{ x \;\big|\; x \subseteq A \big\} 80
  • \big\{ x \;\big|\; x \subset A \big\} 81
  • \big\{ x \;\big|\; x \subseteq A \land x \subseteq B \big\} 82
  • \big\{ x \;\big|\; x \subseteq (A \cap B) \big\} 83
  • \big\{ x \;\big|\; x \subseteq A \lor x \subseteq B \big\} 84
  • \big\{ x \;\big|\; x \subseteq (A \cup B) \big\} 85
  • \Big\{ \mathcal P\big(\{x\}\big) \;\Big|\; x \in A\Big\} 86
  • \big\{ x \;\big|\; x \notin A \big\} 87
  • \big\{ x \;\big|\; x \in \mathbb Z \land x \notin A \big\} 88

  1. true↩︎

  2. false↩︎

  3. false↩︎

  4. true↩︎

  5. true↩︎

  6. false↩︎

  7. true↩︎

  8. true↩︎

  9. true↩︎

  10. false↩︎

  11. false↩︎

  12. true↩︎

  13. true↩︎

  14. true↩︎

  15. false↩︎

  16. false↩︎

  17. true↩︎

  18. true↩︎

  19. true↩︎

  20. false↩︎

  21. mostly true, except for 0 divisors↩︎

  22. false↩︎

  23. true↩︎

  24. false↩︎

  25. true↩︎

  26. false↩︎

  27. mostly true, except for 0 divisors↩︎

  28. false↩︎

  29. true↩︎

  30. false↩︎

  31. false↩︎

  32. false↩︎

  33. false. At a minimum, -1 \mod -1 = 0 \notin \mathbb Z^{-}. Also, there are two interpretations of -4 \mod -3; either it is -1 or it is 2. -1 is more common in programming languages, 2 is more common in the mathematics used in encryption.↩︎

  34. false↩︎

  35. true↩︎

  36. true↩︎

  37. true↩︎

  38. mostly true, except for 0 divisors↩︎

  39. mostly true, except for 0 divisors↩︎

  40. false↩︎

  41. false↩︎

  42. false↩︎

  43. true↩︎

  44. true↩︎

  45. false; 1 \mod 1 = 0↩︎

  46. false↩︎

  47. true↩︎

  48. true↩︎

  49. true↩︎

  50. mostly true, except for 0 divisors↩︎

  51. mostly true, except for 0 divisors↩︎

  52. false because \mathbb R contains negative numbers↩︎

  53. \supset↩︎

  54. \supset↩︎

  55. disjoint↩︎

  56. \neq↩︎

  57. disjoint↩︎

  58. =↩︎

  59. \supset↩︎

  60. \supset↩︎

  61. disjoint↩︎

  62. \subset↩︎

  63. disjoint↩︎

  64. \Big\{ \{\}, \big\{\{\}\big\} \Big\}↩︎

  65. \bigg\{ \{\}, \big\{\{\}\big\}, \Big\{\big\{\{\}\big\}\Big\}, \Big\{\{\}, \big\{\{\}\big\}\Big\} \bigg\}↩︎

  66. \big\{25, 0, 1, \emptyset, \{25\}, \{0\}, \{1\}, \{25,0\}, \{25,1\}, \{0,1\}, \{25,0,1\}\big\}↩︎

  67. 2^{90} which is 1,237,940,039,285,380,274,899,124,224↩︎

  68. 0↩︎

  69. 2^{90}+90 which is 1,237,940,039,285,380,274,899,124,314↩︎

  70. \{1,2,3\}↩︎

  71. \{0,1,2\}↩︎

  72. \{2,3\}↩︎

  73. \{1,2,3,5\}↩︎

  74. \{1\}↩︎

  75. \{2,3,4\}↩︎

  76. \{3,4,5,6,7,8\}↩︎

  77. \big\{ \{1\}, \{2\}, \{3\} \big\}↩︎

  78. \big\{\{1,2\}, \{1,3\}, \{1,5\}, \{2,3\}, \{2,5\}, \{3,5\}\big\}↩︎

  79. \big\{\{1,2\}, \{1,3\}, \{1,5\}, \{2\}, \{2,3\}, \{2,5\}, \{3\}, \{3,5\}\big\}↩︎

  80. \big\{ \{\}, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\} \big\}↩︎

  81. \big\{ \{\}, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\} \big\}↩︎

  82. \big\{ \{\}, \{2\}, \{3\}, \{2,3\}, \big\}↩︎

  83. \big\{ \{\}, \{2\}, \{3\}, \{2,3\}, \big\}↩︎

  84. \big\{ \{\}, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}, \{5\}, \{2,5\}, \{3,5\}, \{2,3,5\} \big\}↩︎

  85. \big\{ \{\}, \{1\}, \{2\}, \{3\}, \{5\}, \{1,2\}, \{1,3\}, \{1,5\}, \{2,3\}, \{2,5\}, \{3,5\}, \{1,2,3\}, \{1,2,5\}, \{1,3,5\}, \{2,3,5\}, \{1,2,3,5\}, \big\}↩︎

  86. \Big\{ \big\{\{\},\{1\}\big\}, \big\{\{\},\{2\}\big\}, \big\{\{\},\{3\}\big\} \Big\}↩︎

  87. An ill-defined set; as written, would contain everything except 1, 2, and 3 but everything is not a mathematically valid concept.↩︎

  88. all integers except 1, 2, and 3. Roughly, \{\dots, -3, -2, -1, 0, 4, 5, 6, 7, \dots\}, though \dots is not a mathematically rigorous symbol.↩︎