### All Introduction to Analysis Resources

## 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 .

**Possible Answers:**

Multiplicative Property

Existence of Multiplicative Identity

Transitive Property

Trichotomy Property

Distributive Law

**Correct answer:**

Trichotomy Property

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

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

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

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

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

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

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

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

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

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.