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Introduction to Analysis : Ordered Field and Completeness Axioms

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Example Question #1 :Ordered Field And Completeness Axioms

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 a<b , b<a , or a=b .

Possible Answers:

Multiplicative Property

Existence of Multiplicative Identity

Transitive Property

三分法的财产

Distributive Law

Correct answer:

三分法的财产

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.

三分法的财产:

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

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For,, and $c\ \epsilon\ \mathbb{R}$ where a0 then this implies ac<bc and a<b and c<0 then this implies bc<ac .

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

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Example Question #2 :Ordered Field And Completeness Axioms

Identify the following property.

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

Possible Answers:

Distribution Laws

三分法的财产

Additive Property

Multiplicative Properties

Transitive Property

Correct answer:

Transitive Property

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.

三分法的财产:

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

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For,, and $c\ \epsilon\ \mathbb{R}$ where a0 then this implies ac<bc and a<b and c<0 then this implies bc<ac .

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

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Example Question #3 :Ordered Field And Completeness Axioms

Identify the following property.

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

Possible Answers:

Multiplicative Properties

Transitive Property

Additive Property

Distribution Laws

三分法的财产

Correct answer:

Additive Property

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.

三分法的财产:

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

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For,, and $c\ \epsilon\ \mathbb{R}$ where a0 then this implies ac<bc and a<b and c<0 then this implies bc<ac .

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

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Example Question #4 :Ordered Field And Completeness Axioms

Identify the following property.

For,, and $c\ \epsilon\ \mathbb{R}$ where a0 then this implies ac<bc and a<b and c<0 then this implies bc<ac .

Possible Answers:

Additive Property

Distribution Laws

Multiplicative Properties

三分法的财产

Transitive Property

Correct answer:

Multiplicative Properties

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.

三分法的财产:

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

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For,, and $c\ \epsilon\ \mathbb{R}$ where a0 then this implies ac<bc and a<b and c<0 then this implies bc<ac .

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

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Introduction to Analysis : Ordered Field and Completeness Axioms

Study concepts, example questions & explanations for Introduction to Analysis

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