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Math 4 axioms on the set of real numbers

L
Leo Crisologo

The document discusses the axioms that define a field and the set of real numbers. It outlines the closure, associativity, commutativity, distributive property, identity element, and inverse element axioms for fields. It also covers equality axioms and proves theorems like cancellation for addition and the involution property using the field axioms.

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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: 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 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 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 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 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 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 and multiplication: 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 and multiplication: 1 Z+ 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 and multiplication: 1 Z+ 2 Z− 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 and multiplication: 1 Z+ 2 Z− 3 {−1, 0, 1} 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 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
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
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
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
Field Axioms: Associativity Associativity Axioms Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 5 / 14
Field Axioms: Associativity Associativity Axioms Addition 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) 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 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 Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 6 / 14
Field Axioms: Commutativity Commutativity Axioms Addition 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 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 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 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 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. 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 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 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 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 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 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 1 1 ∀ a ∈ R − {0}, ∃! a : a· a =1 Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 9 / 14
Equality Axioms Equality Axioms Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 10 / 14
Equality Axioms Equality Axioms 1 Reflexivity: ∀ a ∈ R : a = a 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 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 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 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 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 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 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) 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) 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) 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 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) 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 = 0 : ac = bc → a = b 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 = 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
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
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
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
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 Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 14 / 14
Order Axioms Order Axioms: Inequalities 1 Transitivity for Inequalities 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 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 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 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 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 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 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 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) 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 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) 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 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 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 1 ∀ a ∈ R: a > 0 → > 0 a Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14

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Math 4 axioms on the set of real numbers

  • 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