Ordered Field and Completeness Axioms - Introduction to Analysis

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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 .

Answer

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.

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Question

Identify the following property.

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

Answer

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.

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Question

Identify the following property.

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

Answer

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.

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Question

Identify the following property.

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

Answer

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.

Compare your answer with the correct one above

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