Math 4 axioms on the set of real numbers

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

Field Axioms

Fields

Closure Axioms

Field Axioms

Fields

Closure Axioms

Field Axioms

Fields

Closure Axioms

Field Axioms

Fields

Closure Axioms

Field Axioms

Fields

Closure Axioms

Field Axioms

Fields

Closure Axioms

Field Axioms: Closure

Closure Axioms

Addition: ∀ a, b ∈ R : (a+ b) ∈ R.Multiplication: ∀ a, b ∈ R, (a · b) ∈ R.

Identify if the following sets are closed under addition andmultiplication:

2 Z−

3 {−1, 0, 1}4 {2, 4, 6, 8, 10, ...}5 {−2,−1, 0, 1, 2, 3, ...}6 Q′

2 Z−

3 {−1, 0, 1}4 {2, 4, 6, 8, 10, ...}5 {−2,−1, 0, 1, 2, 3, ...}6 Q′

2 Z−

3 {−1, 0, 1}4 {2, 4, 6, 8, 10, ...}5 {−2,−1, 0, 1, 2, 3, ...}6 Q′

2 Z−

3 {−1, 0, 1}

4 {2, 4, 6, 8, 10, ...}5 {−2,−1, 0, 1, 2, 3, ...}6 Q′

2 Z−

3 {−1, 0, 1}4 {2, 4, 6, 8, 10, ...}

5 {−2,−1, 0, 1, 2, 3, ...}6 Q′

2 Z−

3 {−1, 0, 1}4 {2, 4, 6, 8, 10, ...}5 {−2,−1, 0, 1, 2, 3, ...}

6 Q′

2 Z−

3 {−1, 0, 1}4 {2, 4, 6, 8, 10, ...}5 {−2,−1, 0, 1, 2, 3, ...}6 Q′

2 Z−

3 {−1, 0, 1}4 {2, 4, 6, 8, 10, ...}5 {−2,−1, 0, 1, 2, 3, ...}6 Q′

Field Axioms: Associativity

Addition

∀ a, b, c ∈ R, (a+ b) + c = a+ (b+ c)

Multiplication

∀ a, b, c ∈ R, (a · b) · c = a · (b · c)

Addition

∀ a, b, c ∈ R, (a+ b) + c = a+ (b+ c)

Multiplication

∀ a, b, c ∈ R, (a · b) · c = a · (b · c)

Addition

∀ a, b, c ∈ R, (a+ b) + c = a+ (b+ c)

Multiplication

∀ a, b, c ∈ R, (a · b) · c = a · (b · c)

Addition

∀ a, b, c ∈ R, (a+ b) + c = a+ (b+ c)

Multiplication

∀ a, b, c ∈ R, (a · b) · c = a · (b · c)

Addition

∀ a, b, c ∈ R, (a+ b) + c = a+ (b+ c)

Multiplication

∀ a, b, c ∈ R, (a · b) · c = a · (b · c)

Field Axioms: Commutativity

Addition

∀ a, b ∈ R, a+ b = b+ a

Multiplication

∀ a, b ∈ R, a · b = b · a

Addition

∀ a, b ∈ R, a+ b = b+ a

Multiplication

∀ a, b ∈ R, a · b = b · a

Addition

∀ a, b ∈ R, a+ b = b+ a

Multiplication

∀ a, b ∈ R, a · b = b · a

Addition

∀ a, b ∈ R, a+ b = b+ a

Multiplication

∀ a, b ∈ R, a · b = b · a

Addition

∀ a, b ∈ R, a+ b = b+ a

Multiplication

∀ a, b ∈ R, a · b = b · a

Field Axioms: DPMA

∀ a, b, c ∈ R, c · (a+ b) = c · a+ c · b

Field Axioms: 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.

Addition

∃! 0 : a+ 0 = a for a ∈ R.

Multiplication

∃! 1 : a · 1 = a and 1 · a = a for a ∈ R.

Addition

∃! 0 : a+ 0 = a for a ∈ R.

Multiplication

∃! 1 : a · 1 = a and 1 · a = a for a ∈ R.

Addition

∃! 0 : a+ 0 = a for a ∈ R.

Multiplication

∃! 1 : a · 1 = a and 1 · a = a for a ∈ R.

Addition

∃! 0 : a+ 0 = a for a ∈ R.

Multiplication

∃! 1 : a · 1 = a and 1 · a = a for a ∈ R.

Field Axioms: Existence of an Inverse Element

Addition

∀ a ∈ R,∃! (-a) : a+ (−a) = 0

Multiplication

∀ a ∈ R− {0},∃!(1a

): a · 1a = 1

Addition

∀ a ∈ R,∃! (-a) : a+ (−a) = 0

Multiplication

∀ a ∈ R− {0},∃!(1a

): a · 1a = 1

Addition

∀ a ∈ R,∃! (-a) : a+ (−a) = 0

Multiplication

∀ a ∈ R− {0},∃!(1a

): a · 1a = 1

Addition

∀ a ∈ R,∃! (-a) : a+ (−a) = 0

Multiplication

∀ a ∈ R− {0},∃!(1a

): a · 1a = 1

Addition

∀ a ∈ R,∃! (-a) : a+ (−a) = 0

Multiplication

∀ a ∈ R− {0},∃!(1a

): a · 1a = 1

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

Equality Axioms

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

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

∀ a ∈ R, a 6= 0:1

(1/a)= a

∀ a ∈ R, a 6= 0:1

(1/a)= a

∀ a, b ∈ R : (−a) · b = −(ab)

∀ b ∈ R : (−1) · b = −b (Corollary of previous item)

∀ a ∈ R, a 6= 0:1

(1/a)= a

∀ a ∈ R, a 6= 0:1

(1/a)= a

∀ a ∈ R, a 6= 0:1

(1/a)= a

∀ 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

∀ 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

∀ a ∈ R, a 6= 0:1

(1/a)= a

∀ a ∈ R, a 6= 0:1

(1/a)= a

Order Axioms

Order Axioms: Trichotomy

∀ a, b ∈ R, only one of the following is true:

1 a > b

2 a = b

3 a < b

Order Axioms

1 a > b

2 a = b

3 a < b

Order Axioms

1 a > b

2 a = b

3 a < b

Order Axioms

1 a > b

2 a = b

3 a < b

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

Order Axioms

∀ a, b, c ∈ R : a > b ∧ b > c→ a > c

∀ a, b, c ∈ R : a > b→ a+ c > b+ c

∀ a, b, c ∈ R, c > 0: a > b→ a · c > b · c

Order Axioms

∀ a, b, c ∈ R : a > b ∧ b > c→ a > c

∀ a, b, c ∈ R : a > b→ a+ c > b+ c

∀ a, b, c ∈ R, c > 0: a > b→ a · c > b · c

Order Axioms

∀ a, b, c ∈ R : a > b ∧ b > c→ a > c

∀ a, b, c ∈ R : a > b→ a+ c > b+ c

∀ a, b, c ∈ R, c > 0: a > b→ a · c > b · c

Order Axioms

∀ a, b, c ∈ R : a > b ∧ b > c→ a > c

∀ a, b, c ∈ R : a > b→ a+ c > b+ c

∀ a, b, c ∈ R, c > 0: a > b→ a · c > b · c

Order Axioms

∀ a, b, c ∈ R : a > b ∧ b > c→ a > c

∀ a, b, c ∈ R : a > b→ a+ c > b+ c

∀ a, b, c ∈ R, c > 0: a > b→ a · c > b · c

Order Axioms

∀ a, b, c ∈ R : a > b ∧ b > c→ a > c

∀ a, b, c ∈ R : a > b→ a+ c > b+ c

∀ a, b, c ∈ R, c > 0: a > b→ a · c > b · c

Theorems from the Order Axioms

(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