The Real Number System - Introduction to Analysis

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Question

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 .

Answer

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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Question

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 , , or .

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

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Question

Identify the following property.

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

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

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Question

Identify the following property.

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

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

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Question

Identify the following property.

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

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

Compare your answer with the correct one above

Question

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 .

Answer

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.

Compare your answer with the correct one above

Question

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 , , or .

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

Compare your answer with the correct one above

Question

Identify the following property.

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

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

Compare your answer with the correct one above

Question

Identify the following property.

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

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

Compare your answer with the correct one above

Question

Identify the following property.

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

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

Compare your answer with the correct one above

Question

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 .

Answer

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.

Compare your answer with the correct one above

Question

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 , , or .

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

Compare your answer with the correct one above

Question

Identify the following property.

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

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

Compare your answer with the correct one above

Question

Identify the following property.

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

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

Compare your answer with the correct one above

Question

Identify the following property.

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

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

Compare your answer with the correct one above

Question

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 .

Answer

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.

Compare your answer with the correct one above

Question

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 , , or .

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

Compare your answer with the correct one above

Question

Identify the following property.

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

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

Compare your answer with the correct one above

Question

Identify the following property.

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

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

Compare your answer with the correct one above

Question

Identify the following property.

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

Answer

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 , , or .

Transitive Property:

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

Additive Property:

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

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ 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.

Compare your answer with the correct one above

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