DMT1 – Practice problems with sets
This page does not represent the most current semester of this course; it is present merely as an archive.

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 \big\{x + y \;\big|\; x,y \in \mathbb{Z}^{+} \land x > y \big\}10

  • 3.5 \in \big\{x + y \;\big|\; x \in \mathbb{Z}^{+} \land y \in \mathbb{R}^{+} \big\}11

  • 0 \in \big\{x + y \;\big|\; x,y \in \mathbb{Z}^{+} \land x > y \big\}12

  • 0 \in \big\{x - y \;\big|\; x,y \in \mathbb{R} \land x > y \big\}13

  • 3 \in \{\{1\}, \{2, 3\}, \{4, 5, 6\}\}14

  • \{3\} \in \{\{1\}, \{2, 3\}, \{4, 5, 6\}\}15

  • \{2, 3\} \in \{\{1\}, \{2, 3\}, \{4, 5, 6\}\}16

  • \{2, 3\} \in \mathcal{P}\big(\{2, 3\}\big)17

  • |\{2, 3\}| \in \{2, 3\}18

  • |\{2, 3\}| \in \mathcal{P}\big(\{2, 3\}\big)19

  • \infty \in \mathbb R20

1.2 Qualified membership

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

  • \forall x \in \mathbb R \;.\; x \in \mathbb Q21
  • \forall x \in \mathbb Q \;.\; x \in \mathbb R22
  • \forall x \in \mathbb Z^{+} \;.\; \exists y \in \mathbb Z^{-} \;.\; x + y = 023
  • \forall x \in \mathbb R^{+} \;.\; \exists y \in \mathbb Z^{+} \;.\; 1 \leq \frac{x}{y} \leq 224
  • \exists x \in \mathbb R \;.\; x \in \mathbb N25
  • \exists x \in \mathbb R^{+} \;.\; x \notin \mathbb Q^{+}26
  • \exists x,y \in (\mathbb R \setminus \mathbb N) \;.\; (x \neq y) \land \big((x - y) \in \mathbb N\big)27
  • \forall x \in \mathbb R \;.\; (x \in \mathbb N) \rightarrow (x \in \mathbb Z)28
  • \forall x \in \mathbb Z \;.\; (x \in \mathbb Z^{+}) \lor (x \in \mathbb Z^{-})29
  • \forall x \in \mathbb N \;.\; (x < 0) \rightarrow \bot30
  • \forall x \in \mathbb N \;.\; x \in \big\{ \lfloor y \rfloor \;\big|\; y \in \mathbb R^{+} \big\}31
  • \forall x \in \mathbb N \;.\; x + 1 \in \mathbb N32
  • \forall S \in \{\mathbb Z, \mathbb Q, \mathbb R\}\;.\; \forall x \in S \;.\; x + 1 \in S33
  • \forall x \in \{3, 1, 4, 5\} \;.\; x^{x} \in \{0, 1, 4, 27, 256, 3125, 46656\}34
  • 0 \in \big\{x \;\big|\; \exists y \in \mathbb Z \;.\; y^{y} = x \big\}35
  • \Big|\big\{ x \;\big|\; (x \in \mathbb R) \land (\forall y \in \mathbb N \;.\; x > y) \big\}\Big| \in \{0,1,2\}36
  • 8 \in \big\{x^3 \;\big|\; \exists y \in \mathbb Z \;.\; y^2 = x \big\}37
  • 1 \in \big\{x^3 \;\big|\; \exists y \in \mathbb Z \;.\; y^2 = x \big\}38
  • 64 \in \big\{x^3 \;\big|\; \exists y \in \mathbb Z \;.\; y^2 = x \big\}39

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?
    • addition (+)40
    • subtraction (-)41
    • multiplication (\times)42
    • division (\div)43
    • modulo (\mod{} in math, % in code)44
    • root extraction (\sqrt{})45
  • Which (if any, or all) of the following operators is \mathbb N closed over?
    • addition (+)46
    • subtraction (-)47
    • multiplication (\times)48
    • division (\div)49
    • modulo (\mod{} in math, % in code)50
    • root extraction (\sqrt{})51
  • Which (if any, or all) of the following operators is \mathbb R^{-} closed over?
    • addition (+)52
    • subtraction (-)53
    • multiplication (\times)54
    • division (\div)55
    • modulo (\mod{} in math, % in code)56
    • root extraction (\sqrt{})57
  • Which (if any, or all) of the following operators is \mathbb Q closed over?
    • addition (+)58
    • subtraction (-)59
    • multiplication (\times)60
    • division (\div)61
    • modulo (\mod{} in math, % in code)62
    • root extraction (\sqrt{})63
  • Which (if any, or all) of the following operators is \mathbb Q \setminus \{0\} closed over?
    • addition (+)64
    • subtraction (-)65
    • multiplication (\times)66
    • division (\div)67
    • modulo (\mod{} in math, % in code)68
    • root extraction (\sqrt{})69
  • Which (if any, or all) of the following operators is \mathbb R closed over?
    • addition (+)70
    • subtraction (-)71
    • multiplication (\times)72
    • division (\div)73
    • modulo (\mod{} in math, % in code)74
    • root extraction (\sqrt{})75

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 76 \mathbb Q
\mathbb N 77 \mathbb Z^{+}
even numbers 78 odd numbers
prime numbers 79 odd numbers
\{1, 3, 5\} 80 \{\{1\}, \{3\}, \{5\}\}
\{1, 3, 5\} 81 \{5, 3, 1\}
\{1, 3, 5\} 82 \{5, 3\}
\{0, 1\} 83 \big\{ x \;\big|\; x \in \mathbb{R} \land x^2 = x\big\}
\mathbb{N} 84 \Big\{ x \;\Big|\; x \in \mathbb{R}^{+} \land \big(x - \lfloor x \rfloor = 0\big)\Big\}
even numbers 85 \big\{x \;\big|\; \exists y \in \mathbb Z \;.\; 2y = x\big\}
\mathbb R \setminus \mathbb Z 86 \Big\{ x \;\Big|\; (x \in \mathbb R) \land \big(\forall y \in \mathbb Z \;.\; x \neq y\big) \Big\}
\mathbb R \setminus \mathbb Z 87 \mathbb R \setminus \mathbb Q
\mathbb Q \setminus \mathbb Z 88 \{1, 2, 4\}
\emptyset 89 \mathcal{P}(\emptyset)
\{1\} 90 \mathcal{P}(\{1\})
R^{+} \cup \{0\} 91 \big\{ x \;\big|\; x \in \mathbb R \land \sqrt{x^2} = x \big\}

3 Listing members and cardinality

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

  • \big\{\frac{x}{y} \;\big|\; x\in\{0,1,2\} \land y\in\{1,2,4\} \big\}92
  • \mathcal P \big(\mathcal P(\emptyset)\big)93
  • \mathcal P \Big(\mathcal P \big(\mathcal P(\emptyset)\big)\Big)94
  • \Big\{ x + y \;\Big|\; (x,y \in \mathbb Z) \land (1 < x < y < 10) \land \big(\forall w \in \mathbb Z^{+} \setminus \{1\} \;.\; (x \neq w \rightarrow 0 \neq x \mod{w}) \land (y \neq w \rightarrow 0 \neq y \mod{w}) \big) \Big\}95
  • Assume that A = \{1,2,3,4,5\} and B = \{2,3,5,7\}; \big\{ x \;\big|\; (x \in A) \oplus (x \in B) \big\}96
  • Assume that A = \{25,0,1\}; A \cup \mathcal P(A)97
  • Assume that A is the set of all 2-digit numbers; |\mathcal{P}(A)|98
  • Assume that A is the set of all 2-digit numbers; |\mathcal{P}(A) \cap A|99
  • Assume that A is the set of all 2-digit numbers; |\mathcal{P}(A) \cup A|100

  1. true↩︎

  2. false↩︎

  3. false↩︎

  4. true↩︎

  5. true↩︎

  6. false↩︎

  7. true↩︎

  8. true↩︎

  9. true↩︎

  10. true↩︎

  11. true↩︎

  12. false↩︎

  13. false↩︎

  14. false↩︎

  15. false↩︎

  16. true↩︎

  17. true↩︎

  18. true↩︎

  19. false↩︎

  20. false↩︎

  21. false↩︎

  22. true↩︎

  23. true↩︎

  24. false (consider 0.00001)↩︎

  25. true↩︎

  26. true↩︎

  27. true↩︎

  28. true↩︎

  29. false↩︎

  30. true↩︎

  31. true↩︎

  32. true↩︎

  33. true↩︎

  34. true↩︎

  35. false↩︎

  36. true↩︎

  37. false↩︎

  38. true↩︎

  39. true↩︎

  40. true↩︎

  41. true↩︎

  42. true↩︎

  43. false↩︎

  44. mostly true, except for 0 divisors↩︎

  45. false↩︎

  46. true↩︎

  47. false↩︎

  48. true↩︎

  49. false↩︎

  50. mostly true, except for 0 divisors↩︎

  51. false↩︎

  52. true↩︎

  53. false↩︎

  54. false↩︎

  55. false↩︎

  56. false↩︎

  57. false↩︎

  58. true↩︎

  59. true↩︎

  60. true↩︎

  61. mostly true, except for 0 divisors↩︎

  62. mostly true, except for 0 divisors↩︎

  63. false↩︎

  64. false↩︎

  65. false↩︎

  66. true↩︎

  67. true↩︎

  68. false↩︎

  69. false↩︎

  70. true↩︎

  71. true↩︎

  72. true↩︎

  73. mostly true, except for 0 divisors↩︎

  74. mostly true, except for 0 divisors↩︎

  75. false because \mathbb R contains negative numbers↩︎

  76. \supset↩︎

  77. \supset↩︎

  78. disjoint↩︎

  79. \neq↩︎

  80. disjoint↩︎

  81. =↩︎

  82. \supset↩︎

  83. =↩︎

  84. \supset (would be = if \mathbb Z^{+} instead of \mathbb N↩︎

  85. =↩︎

  86. =↩︎

  87. \supset↩︎

  88. disjoint↩︎

  89. \subset↩︎

  90. disjoint↩︎

  91. =↩︎

  92. \big\{0, \frac{1}{4}, \frac{1}{2}, 1, 2\big\}↩︎

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

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

  95. \{5,7,8,9,10,12\}↩︎

  96. {1, 4, 7}↩︎

  97. {25, 0, 1, , {25}, {0}, {1}, {25,0}, {25,1}, {25,0,1}}↩︎

  98. 2^{90} which is 1,237,940,039,285,380,274,899,124,224↩︎

  99. 0↩︎

  100. 2^{90}+90 which is 1,237,940,039,285,380,274,899,124,314↩︎