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?
• 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?
• 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?
• 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?
• 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?
• 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?
• 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↩︎