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 Element...

# 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. Q' 7. Q ## Field Axioms: Associativity ### Associativity Axioms - ADDITION ∀ a, b, c ∈ R, (a + b) + c = a + (b + c) - MULTIPLICATION ∀ a, b, c∈ R, (ab) c = a (bc) ## 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 ヨ! 0: a + 0 = a for a ∈ R. - MULTIPLICATION ヨ! 1: a1 = 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=1 a ## 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 ## Theorems from the Field and Equality Axioms ### Cancellation for Addition: ∀ a, b, c ∈ R : a + c = b + c → a = c | Statement | Justification | |---|---| | 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 ≠ 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 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 ## 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