All Introduction to Analysis Resources
Example Questions
Example Question #1 :Ordered Field And Completeness Axioms
Identify the following property.
On the spacewhere,only one of the following statements holds true,, or.
Existence of Multiplicative Identity
Transitive Property
Distributive Law
Trichotomy Property
Multiplicative 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,, andwhereandthen this implies.
Additive Property:
For,, andwhereandthen this implies.
Multiplicative Properties:
For,, andwhereandthen this impliesandandthen 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,, andwhereandthen this implies.
Distribution Laws
Trichotomy Property
Transitive Property
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,, andwhereandthen this implies.
Additive Property:
For,, andwhereandthen this implies.
Multiplicative Properties:
For,, andwhereandthen this impliesandandthen this implies.
因此看选项传递公关operty identifies the property in this particular question.
Example Question #3 :Ordered Field And Completeness Axioms
Identify the following property.
For,, andwhereandthen this implies.
Multiplicative Properties
Distribution Laws
Additive Property
Transitive Property
Trichotomy Property
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,, andwhereandthen this implies.
Additive Property:
For,, andwhereandthen this implies.
Multiplicative Properties:
For,, andwhereandthen this impliesandandthen 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,, andwhereandthen this impliesandandthen this implies.
Additive Property
Multiplicative Properties
Distribution Laws
Transitive Property
Trichotomy Property
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,, andwhereandthen this implies.
Additive Property:
For,, andwhereandthen this implies.
Multiplicative Properties:
For,, andwhereandthen this impliesandandthen this implies.
Therefore looking at the options the Multiplicative Properties identifies the property in this particular question.
Example Question #1 :The Real Number System
Determine whether the following statement is true or false:
Ifis a nonempty subset of, thenhas a finite infimum and it is an element of.
True
False
True
According to the Well-Ordered Principal this statement is true. The following proof illuminate its truth.
Supposeis nonempty. From there, it is known thatis bounded above, by.
Therefore, by the Completeness Axiom the supremum ofexists.
Furthermore, ifhas a supremum, then, thus in this particular case.
Thus by the Reflection Principal,
exists and
.
Therefore proving the statement in question true.