math 4 axioms on the set of real numbers
TRANSCRIPT
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Axioms on the Set of Real Numbers
Mathematics 4
June 7, 2011
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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
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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
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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
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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
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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
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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
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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
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Field Axioms: Closure
Closure Axioms
Addition: ∀ a, b ∈ R : (a+ b) ∈ R.Multiplication: ∀ a, b ∈ R, (a · b) ∈ R.
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Field Axioms: Closure
Identify if the following sets are closed under addition andmultiplication:
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
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Field Axioms: Closure
Identify if the following sets are closed under addition andmultiplication:
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
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Field Axioms: Closure
Identify if the following sets are closed under addition andmultiplication:
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
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Field Axioms: Closure
Identify if the following sets are closed under addition andmultiplication:
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
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Field Axioms: Closure
Identify if the following sets are closed under addition andmultiplication:
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
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Field Axioms: Closure
Identify if the following sets are closed under addition andmultiplication:
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
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Field Axioms: Closure
Identify if the following sets are closed under addition andmultiplication:
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
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Field Axioms: Closure
Identify if the following sets are closed under addition andmultiplication:
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
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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)
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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)
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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)
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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)
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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)
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Field Axioms: Commutativity
Commutativity Axioms
Addition
∀ a, b ∈ R, a+ b = b+ a
Multiplication
∀ a, b ∈ R, a · b = b · a
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Field Axioms: Commutativity
Commutativity Axioms
Addition
∀ a, b ∈ R, a+ b = b+ a
Multiplication
∀ a, b ∈ R, a · b = b · a
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Field Axioms: Commutativity
Commutativity Axioms
Addition
∀ a, b ∈ R, a+ b = b+ a
Multiplication
∀ a, b ∈ R, a · b = b · a
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Field Axioms: Commutativity
Commutativity Axioms
Addition
∀ a, b ∈ R, a+ b = b+ a
Multiplication
∀ a, b ∈ R, a · b = b · a
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Field Axioms: Commutativity
Commutativity Axioms
Addition
∀ a, b ∈ R, a+ b = b+ a
Multiplication
∀ a, b ∈ R, a · b = b · a
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Field Axioms: DPMA
Distributive Property of Multiplication over Addition
∀ a, b, c ∈ R, c · (a+ b) = c · a+ c · b
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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.
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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.
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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.
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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.
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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.
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Field Axioms: Existence of an Inverse Element
Existence of an Inverse Element
Addition
∀ a ∈ R,∃! (-a) : a+ (−a) = 0
Multiplication
∀ a ∈ R− {0},∃!(1a
): a · 1a = 1
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Field Axioms: Existence of an Inverse Element
Existence of an Inverse Element
Addition
∀ a ∈ R,∃! (-a) : a+ (−a) = 0
Multiplication
∀ a ∈ R− {0},∃!(1a
): a · 1a = 1
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Field Axioms: Existence of an Inverse Element
Existence of an Inverse Element
Addition
∀ a ∈ R,∃! (-a) : a+ (−a) = 0
Multiplication
∀ a ∈ R− {0},∃!(1a
): a · 1a = 1
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Field Axioms: Existence of an Inverse Element
Existence of an Inverse Element
Addition
∀ a ∈ R,∃! (-a) : a+ (−a) = 0
Multiplication
∀ a ∈ R− {0},∃!(1a
): a · 1a = 1
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Field Axioms: Existence of an Inverse Element
Existence of an Inverse Element
Addition
∀ a ∈ R,∃! (-a) : a+ (−a) = 0
Multiplication
∀ a ∈ R− {0},∃!(1a
): a · 1a = 1
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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
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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
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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
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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
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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
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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
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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
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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 6= 0: ac = bc→ a = b
∀ a ∈ R, a 6= 0:1
(1/a)= a
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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 6= 0: ac = bc→ a = b
∀ a ∈ R, a 6= 0:1
(1/a)= a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14
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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 6= 0: ac = bc→ a = b
∀ a ∈ R, a 6= 0:1
(1/a)= a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14
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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 6= 0: ac = bc→ a = b
∀ a ∈ R, a 6= 0:1
(1/a)= a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14
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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 6= 0: ac = bc→ a = b
∀ a ∈ R, a 6= 0:1
(1/a)= a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14
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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 6= 0: ac = bc→ a = b
∀ a ∈ R, a 6= 0:1
(1/a)= a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14
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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 6= 0: ac = bc→ a = b
∀ a ∈ R, a 6= 0:1
(1/a)= a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14
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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 6= 0: ac = bc→ a = b
∀ a ∈ R, a 6= 0:1
(1/a)= a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14
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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 6= 0: ac = bc→ a = b
∀ a ∈ R, a 6= 0:1
(1/a)= a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14
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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 6= 0: ac = bc→ a = b
∀ a ∈ R, a 6= 0:1
(1/a)= a
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14
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Order Axioms
Order Axioms: Trichotomy
∀ a, b ∈ R, only one of the following is true:
1 a > b
2 a = b
3 a < b
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
∀ a ∈ R : a > 0→ 1
a> 0
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14
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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
∀ a ∈ R : a > 0→ 1
a> 0
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14
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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
∀ a ∈ R : a > 0→ 1
a> 0
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14
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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
∀ a ∈ R : a > 0→ 1
a> 0
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14
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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
∀ a ∈ R : a > 0→ 1
a> 0
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14
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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
∀ a ∈ R : a > 0→ 1
a> 0
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14
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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
∀ a ∈ R : a > 0→ 1
a> 0
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14
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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
∀ a ∈ R : a > 0→ 1
a> 0
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14
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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
∀ a ∈ R : a > 0→ 1
a> 0
Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14