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Axioms on the Set of Real Numbers Mathematics 4 June 7, 2011Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 1 / 14
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Field AxiomsFieldsA ﬁeld is a set where the following axioms hold: Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 2 / 14
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Field AxiomsFieldsA 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
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Field AxiomsFieldsA 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
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Field AxiomsFieldsA 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
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Field AxiomsFieldsA 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
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Field AxiomsFieldsA 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
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Field AxiomsFieldsA 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: ClosureClosure AxiomsAddition: ∀ 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
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Field Axioms: ClosureIdentify if the following sets are closed under addition andmultiplication: Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 4 / 14
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Field Axioms: ClosureIdentify if the following sets are closed under addition andmultiplication: 1 Z+ Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 4 / 14
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Field Axioms: ClosureIdentify if the following sets are closed under addition andmultiplication: 1 Z+ 2 Z− Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 4 / 14
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Field Axioms: ClosureIdentify if the following sets are closed under addition andmultiplication: 1 Z+ 2 Z− 3 {−1, 0, 1} Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 4 / 14
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Field Axioms: ClosureIdentify if the following sets are closed under addition andmultiplication: 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
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Field Axioms: ClosureIdentify 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, ...} Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 4 / 14
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Field Axioms: ClosureIdentify 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 Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 4 / 14
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Field Axioms: ClosureIdentify 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
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Field Axioms: AssociativityAssociativity Axioms Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 5 / 14
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Field Axioms: AssociativityAssociativity Axioms Addition Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 5 / 14
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Field Axioms: AssociativityAssociativity 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
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Field Axioms: AssociativityAssociativity 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
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Field Axioms: AssociativityAssociativity 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
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Field Axioms: CommutativityCommutativity Axioms Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 6 / 14
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Field Axioms: CommutativityCommutativity Axioms Addition Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 6 / 14
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Field Axioms: CommutativityCommutativity Axioms Addition ∀ a, b ∈ R, a + b = b + a Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 6 / 14
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Field Axioms: CommutativityCommutativity Axioms Addition ∀ a, b ∈ R, a + b = b + a Multiplication Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 6 / 14
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Field Axioms: CommutativityCommutativity 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
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Field Axioms: DPMADistributive 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
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Field Axioms: Existence of an Identity ElementExistence of an Identity Element Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 8 / 14
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Field Axioms: Existence of an Identity ElementExistence of an Identity Element Addition Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 8 / 14
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Field Axioms: Existence of an Identity ElementExistence 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
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Field Axioms: Existence of an Identity ElementExistence 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
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Field Axioms: Existence of an Identity ElementExistence 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
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Field Axioms: Existence of an Inverse ElementExistence of an Inverse Element Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 9 / 14
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Field Axioms: Existence of an Inverse ElementExistence of an Inverse Element Addition Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 9 / 14
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Field Axioms: Existence of an Inverse ElementExistence 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
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Field Axioms: Existence of an Inverse ElementExistence 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
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Field Axioms: Existence of an Inverse ElementExistence 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
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Equality AxiomsEquality Axioms Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 10 / 14
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Equality AxiomsEquality Axioms 1 Reﬂexivity: ∀ a ∈ R : a = a Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 10 / 14
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Equality AxiomsEquality 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
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Equality AxiomsEquality 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
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Equality AxiomsEquality 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
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Equality AxiomsEquality 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
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Theorems from the Field and Equality AxiomsCancellation 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
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Theorems from the Field and Equality AxiomsProve the following theorems Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 12 / 14
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Theorems from the Field and Equality AxiomsProve the following theorems Involution: ∀ a ∈ R : − (−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 AxiomsProve 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
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Theorems from the Field and Equality AxiomsProve 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
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Theorems from the Field and Equality AxiomsProve 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
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Theorems from the Field and Equality AxiomsProve 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
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Theorems from the Field and Equality AxiomsProve 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
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Theorems from the Field and Equality AxiomsProve 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
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Theorems from the Field and Equality AxiomsProve 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
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Theorems from the Field and Equality AxiomsProve 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
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Order AxiomsOrder 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
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Order AxiomsOrder 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
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Order AxiomsOrder 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
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Order AxiomsOrder 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 AxiomsOrder Axioms: Inequalities Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 14 / 14
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Order AxiomsOrder Axioms: Inequalities 1 Transitivity for Inequalities Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 14 / 14
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Order AxiomsOrder 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
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Order AxiomsOrder 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
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Order AxiomsOrder 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
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Order AxiomsOrder 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
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Order AxiomsOrder 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 AxiomsProve the following theorems Mathematics 4 () Axioms on the Set of Real Numbers June 7, 2011 15 / 14
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Theorems from the Order AxiomsProve 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
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Theorems from the Order AxiomsProve 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
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Theorems from the Order AxiomsProve 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
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Theorems from the Order AxiomsProve 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
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Theorems from the Order AxiomsProve 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
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Theorems from the Order AxiomsProve 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
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Theorems from the Order AxiomsProve 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
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Theorems from the Order AxiomsProve 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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