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.

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