# Introduction to Analysis : The Real Number System

## 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

Transitive Property

Distributive Law

Trichotomy Property

Multiplicative 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 #1 : The Real Number System

Identify the following property.

For , and  where  and  then this implies .

Trichotomy Property

Transitive Property

Distribution Laws

Multiplicative Properties

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 .

Multiplicative Properties

Distribution Laws

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 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 .

Multiplicative Properties

Distribution Laws

Transitive Property

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 .

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 Multiplicative Properties identifies the property in this particular question.

### Example Question #1 : The Real Number System

Determine whether the following statement is true or false:

If  is a nonempty subset of , then  has a finite infimum and it is an element of .

True

False

True

Explanation:

According to the Well-Ordered Principal this statement is true. The following proof illuminate its truth.

Suppose  is nonempty. From there, it is known that  is bounded above, by .

Therefore, by the Completeness Axiom the supremum of  exists.

Furthermore, if  has a supremum, then , thus in this particular case .

Thus by the Reflection Principal,

exists and

.

Therefore proving the statement in question true.