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Field Axioms and Equality Axioms Quiz

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Questions and Answers

Which axiom states that for any elements a, b, and c in a field, the equation (a + b) + c = a + (b + c) holds?

Associativity Axiom (correct)

Distributive Property Axiom

Commutativity Axiom

Closure Axiom

Which property allows the rearrangement of operands in addition and multiplication, such that a + b = b + a and a·b = b·a?

Closure Property

Associativity Property

Identity Property

Commutativity Property (correct)

Which of the following correctly represents the Existence of an Inverse Element for addition in a field?

For every a ∈ R, there exists an element such that a·1 = a.

For every a ∈ R, there exists an element such that a·(1/a) = 1.

For every a ∈ R, there exists an element such that a + (−a) = 0. (correct)

For every a ∈ R, there exists an element such that a + 0 = a.

What does the Distributive Property of Multiplication over Addition state?

c·(a + b) = c·a + c·b Signup and view all the answers

Which set is NOT closed under addition based on the Closure Axiom?

Ø (the empty set) Signup and view all the answers

What does the equality axiom of reflexivity state?

For all a in R, a = a. Signup and view all the answers

Which statement demonstrates the Cancellation Law for multiplication?

If ac = bc and c ≠ 0, then a = b. Signup and view all the answers

According to the order axioms, what can be concluded if a > b and b > c?

a > c. Signup and view all the answers

Which of the following statements correctly describes the zero property of multiplication?

For any a in R, a · 0 = 0. Signup and view all the answers

What does the theorem regarding the product of two negative numbers state?

The product of two negative numbers is positive. Signup and view all the answers

What does the Addition Property of Inequality indicate?

If a > b, then a + c > b for any c. Signup and view all the answers

Which property states that if a = b, then a · c = b · c?

Multiplication Property of Equality. Signup and view all the answers

According to the trichotomy axiom, which of the following is true for any two real numbers, a and b?

At least one of a > b, a = b, or a Signup and view all the answers

Study Notes

Field Axioms

Fields are sets where addition, multiplication, and other key properties hold.
Closure Axioms: For any two elements (a and b) in a field (R), their sum (a + b) and product (a · b) must also be in the field.
Associativity Axioms: The order of operation doesn't matter for addition and multiplication.
Commutativity Axioms: The order of operands doesn't matter for both addition and multiplication.
Distributive Property of Multiplication over Addition (DPMA): The product of a number with a sum is equal to the sum of the products of the number with each term in the sum.
Existence of an Identity Element: There exist distinct additive (0) and multiplicative (1) identities.
Existence of an Inverse Element: For every element, there exists an additive and a multiplicative inverse.

Equality Axioms

Reflexivity: Any element is equal to itself.
Symmetry: If one element equals another, then the second element also equals the first.
Transitivity: If one element equals a second, and that second element equals a third, then the first element equals the third.
Addition Property of Equality (APE): Adding the same number to both sides of an equation doesn't change its equality.
Multiplication Property of Equality (MPE): Multiplying both sides of an equation by the same number doesn't change its equality.

Theorems from Field and Equality Axioms

Cancellation for Addition: If adding the same number to both sides of an equation results in the same quantities, then both sides were originally equal.

Order Axioms

Trichotomy: Given two real numbers (a and b), one of the following is true: a is greater than b (a > b), a is equal to b (a = b), or a is less than b (a < b).
Transitivity for Inequalities: If a is greater than b, and b is greater than c, then a is greater than c.
Addition Property of Inequality: Adding the same number to both sides of an inequality doesn't change its direction.
Multiplication Property of Inequality: Multiplying both sides of an inequality by a positive number doesn't change its direction.

Theorems from Order Axioms

Addition Property for Positive Numbers (R+): The sum of two positive numbers is positive.
Multiplication Property for Positive Numbers (R+): The product of two positive numbers is positive.
Inverse of Positive and Negative Numbers: The inverse of a positive number is negative, and the inverse of a negative number is positive.
Square of a Number: The square of any number is either 0 or positive.
Existence of an Order for 1: 1 is greater than 0.
Multiplication by Negative Number: Multiplying both sides of an inequality by a negative number reverses the inequality.
Inverse of a Positive Number: The inverse of a positive number is also positive.

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Description

Test your understanding of the fundamental axioms related to fields, including closure, associativity, and commutativity. Additionally, explore the principles of equality axioms such as reflexivity and symmetry. This quiz will challenge your knowledge of these essential mathematical concepts.