1. Axioms on the Set of Real Numbers
Mathematics 4
June 7, 2011
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 1 / 14

2. Field Axioms
Fields
A field is a set where the following axioms hold:
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 2 / 14

3. Field Axioms
Fields
A field is a set where the following axioms hold:
Closure Axioms
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 2 / 14

4. Field Axioms
Fields
A field is a set where the following axioms hold:
Closure Axioms
Associativity Axioms
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 2 / 14

5. Field Axioms
Fields
A field is a set where the following axioms hold:
Closure Axioms
Associativity Axioms
Commutativity Axioms
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 2 / 14

6. Field Axioms
Fields
A field is a set where the following axioms hold:
Closure Axioms
Associativity Axioms
Commutativity Axioms
Distributive Property of Multiplication over Addition
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 2 / 14

7. Field Axioms
Fields
A field is a set where the following axioms hold:
Closure Axioms
Associativity Axioms
Commutativity Axioms
Distributive Property of Multiplication over Addition
Existence of an Identity Element
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 2 / 14

8. Field Axioms
Fields
A field is a set where the following axioms hold:
Closure Axioms
Associativity Axioms
Commutativity Axioms
Distributive Property of Multiplication over Addition
Existence of an Identity Element
Existence of an Inverse Element
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 2 / 14

9. Field Axioms: Closure
Closure Axioms
Addition: ∀ a, b ∈ R : (a + b) ∈ R.
Multiplication: ∀ a, b ∈ R, (a · b) ∈ R.
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 3 / 14

10. Field Axioms: Closure
Identify if the following sets are closed under addition and
multiplication:
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 4 / 14

11. Field Axioms: Closure
Identify if the following sets are closed under addition and
multiplication:
1 Z+
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 4 / 14

12. Field Axioms: Closure
Identify if the following sets are closed under addition and
multiplication:
1 Z+
2 Z−
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 4 / 14

13. Field Axioms: Closure
Identify if the following sets are closed under addition and
multiplication:
1 Z+
2 Z−
3 {−1, 0, 1}
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 4 / 14

14. Field Axioms: Closure
Identify if the following sets are closed under addition and
multiplication:
1 Z+
2 Z−
3 {−1, 0, 1}
4 {2, 4, 6, 8, 10, ...}
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 4 / 14

15. Field Axioms: Closure
Identify if the following sets are closed under addition and
multiplication:
1 Z+
2 Z−
3 {−1, 0, 1}
4 {2, 4, 6, 8, 10, ...}
5 {−2, −1, 0, 1, 2, 3, ...}
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 4 / 14

16. Field Axioms: Closure
Identify if the following sets are closed under addition and
multiplication:
1 Z+
2 Z−
3 {−1, 0, 1}
4 {2, 4, 6, 8, 10, ...}
5 {−2, −1, 0, 1, 2, 3, ...}
6 Q
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 4 / 14

17. Field Axioms: Closure
Identify if the following sets are closed under addition and
multiplication:
1 Z+
2 Z−
3 {−1, 0, 1}
4 {2, 4, 6, 8, 10, ...}
5 {−2, −1, 0, 1, 2, 3, ...}
6 Q
7 Q
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 4 / 14

18. Field Axioms: Associativity
Associativity Axioms
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 5 / 14

19. Field Axioms: Associativity
Associativity Axioms
Addition
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 5 / 14

20. Field Axioms: Associativity
Associativity Axioms
Addition
∀ a, b, c ∈ R, (a + b) + c = a + (b + c)
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 5 / 14

21. Field Axioms: Associativity
Associativity Axioms
Addition
∀ a, b, c ∈ R, (a + b) + c = a + (b + c)
Multiplication
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 5 / 14

22. Field Axioms: Associativity
Associativity Axioms
Addition
∀ a, b, c ∈ R, (a + b) + c = a + (b + c)
Multiplication
∀ a, b, c ∈ R, (a · b) · c = a · (b · c)
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 5 / 14

23. Field Axioms: Commutativity
Commutativity Axioms
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 6 / 14

24. Field Axioms: Commutativity
Commutativity Axioms
Addition
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 6 / 14

25. Field Axioms: Commutativity
Commutativity Axioms
Addition
∀ a, b ∈ R, a + b = b + a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 6 / 14

26. Field Axioms: Commutativity
Commutativity Axioms
Addition
∀ a, b ∈ R, a + b = b + a
Multiplication
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 6 / 14

27. Field Axioms: Commutativity
Commutativity Axioms
Addition
∀ a, b ∈ R, a + b = b + a
Multiplication
∀ a, b ∈ R, a · b = b · a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 6 / 14

28. Field Axioms: DPMA
Distributive Property of Multiplication over Addition
∀ a, b, c ∈ R, c · (a + b) = c · a + c · b
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 7 / 14

29. Field Axioms: Existence of an Identity Element
Existence of an Identity Element
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 8 / 14

30. Field Axioms: Existence of an Identity Element
Existence of an Identity Element
Addition
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 8 / 14

31. Field Axioms: Existence of an Identity Element
Existence of an Identity Element
Addition
∃! 0 : a + 0 = a for a ∈ R.
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 8 / 14

32. Field Axioms: Existence of an Identity Element
Existence of an Identity Element
Addition
∃! 0 : a + 0 = a for a ∈ R.
Multiplication
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 8 / 14

33. Field Axioms: Existence of an Identity Element
Existence of an Identity Element
Addition
∃! 0 : a + 0 = a for a ∈ R.
Multiplication
∃! 1 : a · 1 = a and 1 · a = a for a ∈ R.
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 8 / 14

34. Field Axioms: Existence of an Inverse Element
Existence of an Inverse Element
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 9 / 14

35. Field Axioms: Existence of an Inverse Element
Existence of an Inverse Element
Addition
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 9 / 14

36. Field Axioms: Existence of an Inverse Element
Existence of an Inverse Element
Addition
∀ a ∈ R, ∃! (-a) : a + (−a) = 0
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 9 / 14

37. Field Axioms: Existence of an Inverse Element
Existence of an Inverse Element
Addition
∀ a ∈ R, ∃! (-a) : a + (−a) = 0
Multiplication
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 9 / 14

38. Field Axioms: Existence of an Inverse Element
Existence of an Inverse Element
Addition
∀ a ∈ R, ∃! (-a) : a + (−a) = 0
Multiplication
1 1
∀ a ∈ R − {0}, ∃! a : a· a =1
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 9 / 14

39. Equality Axioms
Equality Axioms
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 10 / 14

40. Equality Axioms
Equality Axioms
1 Reflexivity: ∀ a ∈ R : a = a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 10 / 14

41. Equality Axioms
Equality Axioms
1 Reflexivity: ∀ a ∈ R : a = a
2 Symmetry: ∀ a, b ∈ R : a = b → b = a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 10 / 14

42. Equality Axioms
Equality Axioms
1 Reflexivity: ∀ a ∈ R : a = a
2 Symmetry: ∀ a, b ∈ R : a = b → b = a
3 Transitivity: ∀ a, b, c ∈ R : a = b ∧ b = c → a = c
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 10 / 14

43. Equality Axioms
Equality Axioms
1 Reflexivity: ∀ a ∈ R : a = a
2 Symmetry: ∀ a, b ∈ R : a = b → b = a
3 Transitivity: ∀ a, b, c ∈ R : a = b ∧ b = c → a = c
4 Addition PE: ∀ a, b, c ∈ R : a = b → a + c = b + c
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 10 / 14

44. Equality Axioms
Equality Axioms
1 Reflexivity: ∀ a ∈ R : a = a
2 Symmetry: ∀ a, b ∈ R : a = b → b = a
3 Transitivity: ∀ a, b, c ∈ R : a = b ∧ b = c → a = c
4 Addition PE: ∀ a, b, c ∈ R : a = b → a + c = b + c
5 Multiplication PE: ∀ a, b, c ∈ R : a = b → a · c = b · c
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 10 / 14

45. Theorems from the Field and Equality Axioms
Cancellation for Addition: ∀ a, b, c ∈ R : a + c = b + c → a = c
a+c=b+c Given
a + c + (−c) = b + c + (−c) APE
a + (c + (−c)) = b + (c + (−c)) APA
a+0=b+0 ∃ additive inverses
a=b ∃ additive identity
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 11 / 14

46. Theorems from the Field and Equality Axioms
Prove the following theorems
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14

47. Theorems from the Field and Equality Axioms
Prove the following theorems
Involution: ∀ a ∈ R : − (−a) = a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14

48. Theorems from the Field and Equality Axioms
Prove the following theorems
Involution: ∀ a ∈ R : − (−a) = a
Zero Property of Multiplication: ∀ a ∈ R : a · 0 = 0
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14

49. Theorems from the Field and Equality Axioms
Prove the following theorems
Involution: ∀ a ∈ R : − (−a) = a
Zero Property of Multiplication: ∀ a ∈ R : a · 0 = 0
∀ a, b ∈ R : (−a) · b = −(ab)
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14

50. Theorems from the Field and Equality Axioms
Prove the following theorems
Involution: ∀ a ∈ R : − (−a) = a
Zero Property of Multiplication: ∀ a ∈ R : a · 0 = 0
∀ a, b ∈ R : (−a) · b = −(ab)
∀ b ∈ R : (−1) · b = −b (Corollary of previous item)
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14

51. Theorems from the Field and Equality Axioms
Prove the following theorems
Involution: ∀ a ∈ R : − (−a) = a
Zero Property of Multiplication: ∀ a ∈ R : a · 0 = 0
∀ a, b ∈ R : (−a) · b = −(ab)
∀ b ∈ R : (−1) · b = −b (Corollary of previous item)
(−1) · (−1) = 1 (Corollary of previous item)
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14

52. Theorems from the Field and Equality Axioms
Prove the following theorems
Involution: ∀ a ∈ R : − (−a) = a
Zero Property of Multiplication: ∀ a ∈ R : a · 0 = 0
∀ a, b ∈ R : (−a) · b = −(ab)
∀ b ∈ R : (−1) · b = −b (Corollary of previous item)
(−1) · (−1) = 1 (Corollary of previous item)
∀ a, b ∈ R : (−a) · (−b) = a · b
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14

53. Theorems from the Field and Equality Axioms
Prove the following theorems
Involution: ∀ a ∈ R : − (−a) = a
Zero Property of Multiplication: ∀ a ∈ R : a · 0 = 0
∀ a, b ∈ R : (−a) · b = −(ab)
∀ b ∈ R : (−1) · b = −b (Corollary of previous item)
(−1) · (−1) = 1 (Corollary of previous item)
∀ a, b ∈ R : (−a) · (−b) = a · b
∀ a, b ∈ R : − (a + b) = (−a) + (−b)
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14

54. Theorems from the Field and Equality Axioms
Prove the following theorems
Involution: ∀ a ∈ R : − (−a) = a
Zero Property of Multiplication: ∀ a ∈ R : a · 0 = 0
∀ a, b ∈ R : (−a) · b = −(ab)
∀ b ∈ R : (−1) · b = −b (Corollary of previous item)
(−1) · (−1) = 1 (Corollary of previous item)
∀ a, b ∈ R : (−a) · (−b) = a · b
∀ a, b ∈ R : − (a + b) = (−a) + (−b)
Cancellation Law for Multiplication:
∀ a, b, c ∈ R, c = 0 : ac = bc → a = b
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14

55. Theorems from the Field and Equality Axioms
Prove the following theorems
Involution: ∀ a ∈ R : − (−a) = a
Zero Property of Multiplication: ∀ a ∈ R : a · 0 = 0
∀ a, b ∈ R : (−a) · b = −(ab)
∀ b ∈ R : (−1) · b = −b (Corollary of previous item)
(−1) · (−1) = 1 (Corollary of previous item)
∀ a, b ∈ R : (−a) · (−b) = a · b
∀ a, b ∈ R : − (a + b) = (−a) + (−b)
Cancellation Law for Multiplication:
∀ a, b, c ∈ R, c = 0 : ac = bc → a = b
1
∀ a ∈ R, a = 0 : =a
(1/a)
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14

56. Order Axioms
Order Axioms: Trichotomy
∀ a, b ∈ R, only one of the following is true:
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 13 / 14

57. Order Axioms
Order Axioms: Trichotomy
∀ a, b ∈ R, only one of the following is true:
1 a>b
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 13 / 14

58. Order Axioms
Order Axioms: Trichotomy
∀ a, b ∈ R, only one of the following is true:
1 a>b
2 a=b
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 13 / 14

59. Order Axioms
Order Axioms: Trichotomy
∀ a, b ∈ R, only one of the following is true:
1 a>b
2 a=b
3 a<b
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 13 / 14

60. Order Axioms
Order Axioms: Inequalities
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 14 / 14

61. Order Axioms
Order Axioms: Inequalities
1 Transitivity for Inequalities
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 14 / 14

62. Order Axioms
Order Axioms: Inequalities
1 Transitivity for Inequalities
∀ a, b, c ∈ R : a > b ∧ b > c → a > c
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 14 / 14

63. Order Axioms
Order Axioms: Inequalities
1 Transitivity for Inequalities
∀ a, b, c ∈ R : a > b ∧ b > c → a > c
2 Addition Property of Inequality
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 14 / 14

64. Order Axioms
Order Axioms: Inequalities
1 Transitivity for Inequalities
∀ a, b, c ∈ R : a > b ∧ b > c → a > c
2 Addition Property of Inequality
∀ a, b, c ∈ R : a > b → a + c > b + c
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 14 / 14

65. Order Axioms
Order Axioms: Inequalities
1 Transitivity for Inequalities
∀ a, b, c ∈ R : a > b ∧ b > c → a > c
2 Addition Property of Inequality
∀ a, b, c ∈ R : a > b → a + c > b + c
3 Multiplication Property of Inequality
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 14 / 14

66. Order Axioms
Order Axioms: Inequalities
1 Transitivity for Inequalities
∀ a, b, c ∈ R : a > b ∧ b > c → a > c
2 Addition Property of Inequality
∀ a, b, c ∈ R : a > b → a + c > b + c
3 Multiplication Property of Inequality
∀ a, b, c ∈ R, c > 0 : a > b → a · c > b · c
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 14 / 14

67. Theorems from the Order Axioms
Prove the following theorems
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14

68. Theorems from the Order Axioms
Prove the following theorems
(4-1) R+ is closed under addition:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a + b > 0
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14

69. Theorems from the Order Axioms
Prove the following theorems
(4-1) R+ is closed under addition:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a + b > 0
(4-2) R+ is closed under multiplication:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a · b > 0
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14

70. Theorems from the Order Axioms
Prove the following theorems
(4-1) R+ is closed under addition:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a + b > 0
(4-2) R+ is closed under multiplication:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a · b > 0
(4-3) ∀ a ∈ R : (a > 0 → −a < 0) ∧ (a < 0 → −a > 0)
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14

71. Theorems from the Order Axioms
Prove the following theorems
(4-1) R+ is closed under addition:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a + b > 0
(4-2) R+ is closed under multiplication:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a · b > 0
(4-3) ∀ a ∈ R : (a > 0 → −a < 0) ∧ (a < 0 → −a > 0)
(4-4) ∀ a, b ∈ R : a > b → −b > −a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14

72. Theorems from the Order Axioms
Prove the following theorems
(4-1) R+ is closed under addition:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a + b > 0
(4-2) R+ is closed under multiplication:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a · b > 0
(4-3) ∀ a ∈ R : (a > 0 → −a < 0) ∧ (a < 0 → −a > 0)
(4-4) ∀ a, b ∈ R : a > b → −b > −a
(4-5) ∀ a ∈ R : (a2 = 0) ∨ (a2 > 0)
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14

73. Theorems from the Order Axioms
Prove the following theorems
(4-1) R+ is closed under addition:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a + b > 0
(4-2) R+ is closed under multiplication:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a · b > 0
(4-3) ∀ a ∈ R : (a > 0 → −a < 0) ∧ (a < 0 → −a > 0)
(4-4) ∀ a, b ∈ R : a > b → −b > −a
(4-5) ∀ a ∈ R : (a2 = 0) ∨ (a2 > 0)
(4-6) 1 > 0
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14

74. Theorems from the Order Axioms
Prove the following theorems
(4-1) R+ is closed under addition:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a + b > 0
(4-2) R+ is closed under multiplication:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a · b > 0
(4-3) ∀ a ∈ R : (a > 0 → −a < 0) ∧ (a < 0 → −a > 0)
(4-4) ∀ a, b ∈ R : a > b → −b > −a
(4-5) ∀ a ∈ R : (a2 = 0) ∨ (a2 > 0)
(4-6) 1 > 0
∀ a, b, c ∈ R : (a > b) ∧ (0 > c) → b · c > a · c
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14

75. Theorems from the Order Axioms
Prove the following theorems
(4-1) R+ is closed under addition:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a + b > 0
(4-2) R+ is closed under multiplication:
∀ a, b ∈ R : a > 0 ∧ b > 0 → a · b > 0
(4-3) ∀ a ∈ R : (a > 0 → −a < 0) ∧ (a < 0 → −a > 0)
(4-4) ∀ a, b ∈ R : a > b → −b > −a
(4-5) ∀ a ∈ R : (a2 = 0) ∨ (a2 > 0)
(4-6) 1 > 0
∀ a, b, c ∈ R : (a > b) ∧ (0 > c) → b · c > a · c
1
∀ a ∈ R: a > 0 → > 0
a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14