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Introduction to Analysis : Intro Analysis

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Example Question #1 : Intro Analysis

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 a<b , b<a , or a=b .

Possible Answers:

Distributive Law

Multiplicative Property

Trichotomy Property

Transitive Property

Existence of Multiplicative Identity

Correct answer:

Trichotomy Property

Explanation:

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 a<b , b<a , or a=b .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ where a0 then this implies ac<bc and a<b and c<0 then this implies bc<ac .

Therefore looking at the options the Trichotomy Property identifies the property in this particular question.

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Example Question #2 : Ordered Field And Completeness Axioms

Identify the following property.

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

Possible Answers:

Additive Property

Transitive Property

Trichotomy Property

Distribution Laws

Multiplicative Properties

Correct answer:

Transitive Property

Explanation:

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 a<b , b<a , or a=b .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ where a0 then this implies ac<bc and a<b and c<0 then this implies bc<ac .

Therefore looking at the options the Transitive Property identifies the property in this particular question.

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Example Question #3 : Ordered Field And Completeness Axioms

Identify the following property.

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

Possible Answers:

Transitive Property

Distribution Laws

Multiplicative Properties

Additive Property

Trichotomy Property

Correct answer:

Additive Property

Explanation:

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 a<b , b<a , or a=b .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ where a0 then this implies ac<bc and a<b and c<0 then this implies bc<ac .

Therefore looking at the options the Additive Property identifies the property in this particular question.

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Example Question #4 : Ordered Field And Completeness Axioms

Identify the following property.

For , , and $c\ \epsilon\ \mathbb{R}$ where a0 then this implies ac<bc and a<b and c<0 then this implies bc<ac .

Possible Answers:

Additive Property

Transitive Property

Multiplicative Properties

Trichotomy Property

Distribution Laws

Correct answer:

Multiplicative Properties

Explanation:

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 a<b , b<a , or a=b .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ where a0 then this implies ac<bc and a<b and c<0 then this implies bc<ac .

Therefore looking at the options the Multiplicative Properties identifies the property in this particular question.

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Example Question #1 : Induction

Determine whether the following statement is true or false:

If is a nonempty subset of $\mathbb{N}$ , 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 $A\subseteq \mathbb{N}$ is nonempty. From there, it is known that is bounded above, by .

Therefore, by the Completeness Axiom the supremum of exists.

Furthermore, if $A\subset \mathbb{Z}$ has a supremum, then $\text{sup}A\ \epsilon\ A$ , thus in this particular case $\text{sup}(-A)\ \epsilon\ -A$ .

Thus by the Reflection Principal,

$\text{inf}A=-\text{sup}(-A)$

exists and

$\text{inf}A\ \epsilon\ -(-A)=A$ .

Therefore proving the statement in question true.

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Example Question #1 : Limits Of Sequences

What term does the following define.

A sequence of sets $\begin{Bmatrix} I_n \end{Bmatrix}_{n\ \epsilon\ \mathbb{N}}$ is __________ if and only if $I_1\supseteq I_2\supseteq ...$ .

Possible Answers:

Increasing

Unbounded

Decreasing

Nested

Bounded

Correct answer:

Nested

Explanation:

This statement:

A sequence of sets $\begin{Bmatrix} I_n \end{Bmatrix}_{n\ \epsilon\ \mathbb{N}}$ is __________ if and only if $I_1\supseteq I_2\supseteq ...$

is the definition of nested.

This means that the sequence I_n 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.

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Example Question #1 : Intro Analysis

Determine whether the following statement is true or false:

Let , , , and $d\ \epsilon\ \mathbb{R}$ . If a<b and c<d<0 then ac<bd .

Possible Answers:

False

True

Correct answer:

False

Explanation:

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

Let

a=1, b=2, c=-3, d=-2

First make sure the inequalities hold true.

$\\a<b \\1<2$

and

$\\c<d<0 \\-3<-2<0$

Now find the products.

$\\ac<bd \\(1)(-3)<(2)(-2) \\-3\nless -4$

Therefore, the statement is false.

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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:

, $b\ \epsilon\ \mathbb{R}$ , a>b , and $f:[a,b]\rightarrow \mathbb{Z}$

, $b\ \epsilon\ \mathbb{R}$ , a>b , and $f:[a,b]\rightarrow \mathbb{R}$

, $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded

, $b\ \epsilon\ \mathbb{Z}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded

, $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{Z}$ be bounded

Correct answer:

, $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ 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 U(f,P) represents the upper integral and L(f,P) the following are defined:

1. The upper integral of on [a,b] is

$(U)\int_a^bf(x)dx:=\text{inf}\begin{Bmatrix} U(f,p)) \end{Bmatrix}$ where is a partition of [a,b] .

2. The lower integral of on [a,b] is

$(L)\int_a^bf(x)dx:=\text{sup}\begin{Bmatrix} L(f,p)) \end{Bmatrix}$ where is a partition of [a,b] .

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

$\int_a^bf(x):=(U)\int_a^bf(x)dx=(L)\int_a^bf(x)dx$

if and only if , $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded.

Therefore the necessary condition for the proving the upper and lower integrals of a bounded function exists is if and only if , $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded.

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Example Question #2 : Riemann Integral, Riemann Sums, & Improper Riemann Integration

What term has the following definition.

, $b\ \epsilon\ \mathbb{R}$ and a<b . Over the interval [a,b] is a set of points $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ such that

a=x_0<x_1<...<x_n=b

Possible Answers:

Upper Riemann sum

Lower Riemann sum

Refinement of a partition

Partition

Norm

Correct answer:

Partition

Explanation:

By definition

If , $b\ \epsilon\ \mathbb{R}$ and a<b .

A partition over the interval [a,b] is a set of points $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ such that

a=x_0<x_1<...<x_n=b .

Therefore, the term that describes this statement is partition.

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

What term has the following definition.

The __________ of a partition $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ is

$||P||=\max_{1\leq j\leq n}|x_j-x_{j-1}|$

Possible Answers:

Norm

Partition

Refinement of a partition

Lower Riemann sum

Upper Riemann Sum

Correct answer:

Norm

Explanation:

By definition

If , $b\ \epsilon\ \mathbb{R}$ and a<b .

A partition over the interval [a,b] is a set of points $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ such that

a=x_0<x_1<...<x_n=b .

Furthermore,

The norm of the partition

$P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ is

$||P||=\max_{1\leq j\leq n}|x_j-x_{j-1}|$

Therefore, the term that describes this statement is norm.

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Introduction to Analysis : Intro Analysis

Study concepts, example questions & explanations for Introduction to Analysis

All Introduction to Analysis Resources

Example Questions

Example Question #1 : Intro Analysis

Example Question #2 : Ordered Field And Completeness Axioms

Example Question #3 : Ordered Field And Completeness Axioms

Example Question #4 : Ordered Field And Completeness Axioms

Example Question #1 : Induction

Example Question #1 : Limits Of Sequences

Example Question #1 : Intro Analysis

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

Example Question #2 : Riemann Integral, Riemann Sums, & Improper Riemann Integration

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

All Introduction to Analysis Resources

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