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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 9 / 14

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

Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 9 / 14

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

Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 9 / 14

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

Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 9 / 14

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

Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 9 / 14

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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