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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
.
Distributive Law
Existence of Multiplicative Identity
Multiplicative Property
Trichotomy Property
Transitive Property
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 : The Real Number System
Identify the following property.
For ,
, and
where
and
then this implies
.
Trichotomy Property
Transitive Property
Distribution Laws
Additive Property
Multiplicative Properties
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 : The Real Number System
Identify the following property.
For ,
, and
where
and
then this implies
.
Transitive Property
Distribution Laws
Additive Property
Trichotomy Property
Multiplicative Properties
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 : The Real Number System
Identify the following property.
For ,
, and
where
and
then this implies
and
and
then this implies
.
Additive Property
Distribution Laws
Transitive Property
Trichotomy Property
Multiplicative Properties
Multiplicative Properties
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 : Intro Analysis
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
.
False
True
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.
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