Introduction to Analysis : Intro Analysis

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Example Questions

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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 .

Possible Answers:

Multiplicative Property

Existence of Multiplicative Identity

Transitive Property

Trichotomy Property

Distributive Law

Correct answer:

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 .

Additive Property:

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 .

Possible Answers:

Distribution Laws

Trichotomy Property

Additive Property

Multiplicative Properties

Transitive Property

Correct answer:

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 .

Additive Property:

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 .

Possible Answers:

Multiplicative Properties

Transitive Property

Additive Property

Distribution Laws

Trichotomy Property

Correct answer:

Additive 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 .

Additive Property:

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 .

Possible Answers:

Additive Property

Distribution Laws

Multiplicative Properties

Trichotomy Property

Transitive Property

Correct answer:

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 .

Additive Property:

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 : Induction

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 .

Possible Answers:

True

False

Correct answer:

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.

Example Question #1 : Limits Of Sequences

What term does the following define.

A sequence of sets  is __________ if and only if .

Possible Answers:

Bounded

Nested

Decreasing

Increasing

Unbounded

Correct answer:

Nested

Explanation:

This statement:

A sequence of sets  is __________ if and only if 

is the definition of nested.

This means that the sequence  for all  elements, for which  belongs to the natural numbers, is considered a nested set if and only if the subsequent sets are subsets of it. 

Other theorems in intro analysis build off this understanding.

Example Question #1 : Differentiability On Real Numbers (R)

Determine whether the following statement is true or false:

Let , and . If  and  then .

Possible Answers:

True

False

Correct answer:

False

Explanation:

Determine this statement is false by showing a contradiction when actual values are used.

Let 

First make sure the inequalities hold true.

 

and

Now find the products.

Therefore, the statement is false.

Example Question #1 : Riemann Integral, Riemann Sums, & Improper Riemann Integration

What conditions are necessary to prove that the upper and lower integrals of a bounded function exist?

Possible Answers:

,  and 

,  and  be bounded

,  and  be bounded

,  and 

,  and  be bounded

Correct answer:

,  and  be bounded

Explanation:

Using the definition for Riemann sums to define the upper and lower integrals of a function answers the question.

According the the Riemann sum where  represents the upper integral and  the following are defined:

1. The upper integral of  on  is

 where  is a partition of .

2. The lower integral of  on  is

where  is a partition of .

3. If 1 and 2 are the same then the integral is said to be

if and only if ,  and  be bounded.

Therefore the necessary condition for the proving the upper and lower integrals of a bounded function exists is if and only if ,  and  be bounded.

Example Question #1 : Intro Analysis

What term has the following definition.

 and . Over the interval  is a set of points  such that

Possible Answers:

Lower Riemann sum

Refinement of a partition

Upper Riemann sum

Norm

Partition

Correct answer:

Partition

Explanation:

By definition 

If  and .

A partition over the interval  is a set of points  such that

.

Therefore, the term that describes this statement is partition.

Example Question #3 : Riemann Integral, Riemann Sums, & Improper Riemann Integration

What term has the following definition.

The __________ of a partition   is

Possible Answers:

Norm

Upper Riemann Sum

Lower Riemann sum

Partition

Refinement of a partition

Correct answer:

Norm

Explanation:

By definition 

If  and .

A partition over the interval  is a set of points  such that

.

Furthermore, 

The norm of the partition 

 is

Therefore, the term that describes this statement is norm.

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