Ordered Field and Completeness Axioms

Introduction to Analysis · Learn by Concept

Help Questions

Introduction to Analysis › Ordered Field and Completeness Axioms

1 - 4

Identify the following property.

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies .

Transitive Property

CORRECT

Trichotomy Property

Additive Property

Multiplicative Properties

Distribution Laws

Explanation

The real number system, $\mathbb{R}\times \mathbb{R}$ contains order axioms that show relations and properties of the system that add completeness to the ordered field algebraic laws.

The properties are as follows.

Trichotomy Property:

Given , $b\ \epsilon\ \mathbb{R}$ only one of the following statements holds true , , or .

Transitive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies .

Additive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and $c\ \epsilon\ \mathbb{R}$ then this implies .

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies and and then this implies .

Therefore looking at the options the Transitive Property identifies the property in this particular question.

Identify the following property.

On the space $\mathbb{R}\times \mathbb{R}$ where , $b\ \epsilon\ \mathbb{R}$ only one of the following statements holds true , , or .

Trichotomy Property

CORRECT

Transitive Property

Multiplicative Property

Existence of Multiplicative Identity

Distributive Law

Explanation

The real number system, $\mathbb{R}\times \mathbb{R}$ contains order axioms that show relations and properties of the system that add completeness to the ordered field algebraic laws.

The properties are as follows.

Trichotomy Property:

Given , $b\ \epsilon\ \mathbb{R}$ only one of the following statements holds true , , or .

Transitive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies .

Additive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and $c\ \epsilon\ \mathbb{R}$ then this implies .

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies and and then this implies .

Therefore looking at the options the Trichotomy Property identifies the property in this particular question.

Identify the following property.

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies and and then this implies .

Multiplicative Properties

CORRECT

Additive Property

Transitive Property

Trichotomy Property

Distribution Laws

Explanation

The real number system, $\mathbb{R}\times \mathbb{R}$ contains order axioms that show relations and properties of the system that add completeness to the ordered field algebraic laws.

The properties are as follows.

Trichotomy Property:

Given , $b\ \epsilon\ \mathbb{R}$ only one of the following statements holds true , , or .

Transitive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies .

Additive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and $c\ \epsilon\ \mathbb{R}$ then this implies .

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies and and then this implies .

Therefore looking at the options the Multiplicative Properties identifies the property in this particular question.

Identify the following property.

For , , and $c\ \epsilon\ \mathbb{R}$ where and $c\ \epsilon\ \mathbb{R}$ then this implies .

Additive Property

CORRECT

Transitive Property

Trichotomy Property

Multiplicative Properties

Distribution Laws

Explanation

The real number system, $\mathbb{R}\times \mathbb{R}$ contains order axioms that show relations and properties of the system that add completeness to the ordered field algebraic laws.

The properties are as follows.

Trichotomy Property:

Given , $b\ \epsilon\ \mathbb{R}$ only one of the following statements holds true , , or .

Transitive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies .

Additive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and $c\ \epsilon\ \mathbb{R}$ then this implies .

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies and and then this implies .

Therefore looking at the options the Additive Property identifies the property in this particular question.

Return to subject

AI Study CoachYour personal study buddy

Instant TutoringConnect with a tutor now

Request a TutorGet matched with an expert

Live ClassesJoin interactive classes

Question of the DayDaily practice to build your skills

FlashcardsStudy and memorize key concepts

WorksheetsBuild custom practice quizzes

AI SolverStep-by-step problem solutions

Learn by ConceptMaster concepts step by step

Diagnostic TestsAssess your knowledge level

Practice TestsTest your skills with exam questions

GamesLearn through free games