# Introduction to Analysis : Ordered Field and Completeness Axioms

## Example Questions

### Example Question #1 : Ordered Field And Completeness Axioms

Identify the following property.

On the space  where  only one of the following statements holds true , or .

Existence of Multiplicative Identity

Trichotomy Property

Distributive Law

Multiplicative Property

Transitive Property

Trichotomy Property

Explanation:

The real number system,  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  only one of the following statements holds true , or .

Transitive Property:

For , and  where  and  then this implies .

For , and  where  and  then this implies .

Multiplicative Properties:

For , and  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.

### Example Question #2 : Ordered Field And Completeness Axioms

Identify the following property.

For , and  where  and  then this implies .

Multiplicative Properties

Distribution Laws

Transitive Property

Trichotomy Property

Transitive Property

Explanation:

The real number system,  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  only one of the following statements holds true , or .

Transitive Property:

For , and  where  and  then this implies .

For , and  where  and  then this implies .

Multiplicative Properties:

For , and  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.

### Example Question #3 : Ordered Field And Completeness Axioms

Identify the following property.

For , and  where  and  then this implies .

Transitive Property

Trichotomy Property

Distribution Laws

Multiplicative Properties

Explanation:

The real number system,  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  only one of the following statements holds true , or .

Transitive Property:

For , and  where  and  then this implies .

For , and  where  and  then this implies .

Multiplicative Properties:

For , and  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.

### Example Question #4 : Ordered Field And Completeness Axioms

Identify the following property.

For , and  where  and  then this implies  and   and  then this implies .

Transitive Property

Distribution Laws

Multiplicative Properties

Trichotomy Property

Multiplicative Properties

Explanation:

The real number system,  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  only one of the following statements holds true , or .

Transitive Property:

For , and  where  and  then this implies .