1.2 Axioms on the Set of Real Numbers.pdf

# Axioms on the Set of Real Numbers ## Mathematics 4 June 7, 2011 ## 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 Elem...

# Axioms on the Set of Real Numbers ## Mathematics 4 June 7, 2011 ## 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: 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 and multiplication: 1. Z+ 2. Z− 3. {−1, 0, 1} 4. {2, 4, 6, 8, 10,…} 5. {−2, −1, 0, 1, 2, 3,…} 6. Ø 7. Q ## 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) ## Field Axioms: Commutativity ### Commutativity Axioms * **ADDITION** ∀ a, b ∈ R, a + b = b + a * **MULTIPLICATION** ∀ a, b ∈ R, a·b = b·a ## Field Axioms: DPMA ### Distributive Property of Multiplication over Addition ∀ a, b, c ∈ R, c·(a + b) = c·a + c·b ## 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. ## Field Axioms: Existence of an Inverse Element ### Existence of an Inverse Element * **ADDITION** ∀ a ∈ R, ∃! (−a): a + (−a) = 0 * **MULTIPLICATION** ∀ a ∈ R − {0}, ∃! (1/a): a·(1/a) = 1 ## Equality Axioms ### Equality Axioms * **Reflexivity**: ∀ a ∈ R: a = a * **Symmetry**: ∀ a, b∈R: a = b → b = a * **Transitivity**: ∀ a, b, c∈R: a = b ∧ b = c → a = c * **Addition PE**: ∀ a, b, c ∈ R: a = b → a + c = b + c * **Multiplication PE**: ∀ a, b, c ∈ R: a = b → a·c = b·c ## Theorems from the Field and Equality Axioms ### Cancellation for Addition: ∀ a, b, c ∈ R: a + c = b + c → a = c | Step | Description | |---|---| | 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 = −(a·b) * ∀ 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 * ∀ a ∈ R, a ≠ 0: (1/a) = 1/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: Inequalities * **Transitivity for Inequalities** ∀ a, b, c ∈ R: a > b ∧ b > c → a > c * **Addition Property of Inequality** ∀ a, b, c ∈ R: a > b → a + c > b + c * **Multiplication Property of Inequality** ∀ a, b, c ∈ R, c > 0: a > b → a · c > b · c ## 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) V (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.