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

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

• $\infty \in \mathbb R$¹⁶

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?
  • addition ($+$)¹⁷
  • subtraction ($-$)¹⁸
  • multiplication ($\times$)¹⁹
  • division ($\div$)²⁰
  • modulo ($\mod{}$ in math, % in code)²¹
  • root extraction ($\sqrt{}$)²²
• Which (if any, or all) of the following operators is $\mathbb N$ closed over?
  • addition ($+$)²³
  • subtraction ($-$)²⁴
  • multiplication ($\times$)²⁵
  • division ($\div$)²⁶
  • modulo ($\mod{}$ in math, % in code)²⁷
  • root extraction ($\sqrt{}$)²⁸
• Which (if any, or all) of the following operators is $\mathbb R^{-}$ closed over?
  • addition ($+$)²⁹
  • subtraction ($-$)³⁰
  • multiplication ($\times$)³¹
  • division ($\div$)³²
  • modulo ($\mod{}$ in math, % in code)³³
  • root extraction ($\sqrt{}$)³⁴
• Which (if any, or all) of the following operators is $\mathbb Q$ closed over?
  • addition ($+$)³⁵
  • subtraction ($-$)³⁶
  • multiplication ($\times$)³⁷
  • division ($\div$)³⁸
  • modulo ($\mod{}$ in math, % in code)³⁹
  • root extraction ($\sqrt{}$)⁴⁰
• Which (if any, or all) of the following operators is $\mathbb Q \setminus \{0\}$ closed over?
  • addition ($+$)⁴¹
  • subtraction ($-$)⁴²
  • multiplication ($\times$)⁴³
  • division ($\div$)⁴⁴
  • modulo ($\mod{}$ in math, % in code)⁴⁵
  • root extraction ($\sqrt{}$)⁴⁶
• Which (if any, or all) of the following operators is $\mathbb R$ closed over?
  • addition ($+$)⁴⁷
  • subtraction ($-$)⁴⁸
  • multiplication ($\times$)⁴⁹
  • division ($\div$)⁵⁰
  • modulo ($\mod{}$ in math, % in code)⁵¹
  • root extraction ($\sqrt{}$)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

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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.↩︎