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?

• $3 \in \mathbb Z$¹

• $3.5 \in \mathbb Z$²

• $\pi \in \mathbb Z$³

• $3 \in \mathbb Q$⁴

• $3.5 \in \mathbb Q$⁵

• $\pi \in \mathbb Q$⁶

• $3 \in \mathbb R$⁷

• $3.5 \in \mathbb R$⁸

• $\pi \in \mathbb R$⁹

• $3 \in \big\{x + y \;\big|\; x,y \in \mathbb{Z}^{+} \land x > y \big\}$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

• $\infty \in \mathbb R$²⁰

1.2 Qualified membership

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

• $\forall x \in \mathbb R \;.\; x \in \mathbb Q$²¹
• $\forall x \in \mathbb Q \;.\; x \in \mathbb R$²²
• $\forall x \in \mathbb Z^{+} \;.\; \exists y \in \mathbb Z^{-} \;.\; x + y = 0$²³
• $\forall x \in \mathbb R^{+} \;.\; \exists y \in \mathbb Z^{+} \;.\; 1 \leq \frac{x}{y} \leq 2$²⁴
• $\exists x \in \mathbb R \;.\; x \in \mathbb N$²⁵
• $\exists x \in \mathbb R^{+} \;.\; x \notin \mathbb Q^{+}$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
• $\forall x \in \mathbb N \;.\; (x < 0) \rightarrow \bot$³⁰
• $\forall x \in \mathbb N \;.\; x \in \big\{ \lfloor y \rfloor \;\big|\; y \in \mathbb R^{+} \big\}$³¹
• $\forall x \in \mathbb N \;.\; x + 1 \in \mathbb N$³²
• $\forall S \in \{\mathbb Z, \mathbb Q, \mathbb R\}\;.\; \forall x \in S \;.\; x + 1 \in S$³³
• $\forall x \in \{3, 1, 4, 5\} \;.\; x^{x} \in \{0, 1, 4, 27, 256, 3125, 46656\}$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

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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↩︎