Calculus & Classical Analysis

This post is from a book - Introduction to Calculus and Classical Analysis 3ed , 2011 and the author is Mar Hijab. In this post , I extract some topics and examples from it and post them here. Basically , those are some common concepts in mathematics analysis and calculus. Anyway , we start to discuss them.
The Set of Real Numbers 
 We may assume a set R , it is a set which consists of infinite real numbers .  What is a real number ? From wiki ,  a real number is a value that represents a quantity along a continuous line. R consists of  two kinds , one is rational numbers  and  the other is irrational numbers. For any rational number , it can be represented as a form , p/q ; p and q are integral numbers and q ≠ 0 .  Because p.q are integers and they are countable and infinite , there are infinite rational numbers. We may conclude a result . R = R' ∪ I , R' is rational numbers set , and I is irrational numbers set. So |R| = ∞ ,  its number is infinite , but they are countable.  Again ,  we start to define the definition of set as below: 
Suppose that there are three sets universe set U , and two sets A and B under set U. For any two sets A and B ,   there are some basic properties as below : 
1. intersection - symbol ∩ , the definition of it as below: 
A ∩ B = { x | x ∈ A and x ∈ B } , ∈ is a symbol that means belongs to  and x is an element .  
2. union - symbol ∪ , the definition of it as below: 
A ∪ B = { x | x ∈ A or x ∈ B   } 
The above are two basic operations between sets , their symbols are standard. 
Next , other operations among them as followings :   
First , we introduce a difference operation - . As the above ,  suppose there are two sets A and B and they are under universe set U. This operation can be defined as below : 
A - B = { x | = x ∈ A , but x ∉ B } , Basically , we can use an easy way ( Venn's Diagram ) to represent it. 

A - B 

That means for any element x belongs to set A , but it doesn't belong to set B . From this property , we may derive some properties .
3. Complement laws - symbol \ , the definition of it as below:
U \ A  = { x | x   ∈  U , but  x ∉ A  } , we may use a symbol ~ to represent it.  
4. symmetric difference , symbol △ , the definition of it as below: 
A △ B = { x |  x ∈ A , but x ∉ B or x ∈ B , but x ∉ A  }  , its Venn diagram as below : 

A △ B

From the above venn diagram , we can use standard operations to represent it.

A △ B  = A ∪ B - A ∩ B  
And other relevant set operations , please refer to some basic set theory books or wiki . 
Of course , we may extend the above elementary set operations as below : 
Assume there are n sets , they are A1, A2, A3, ... , An and they all under Universe Set U . 
An extension principle ( intersection ) 
A1  ∩ A2  ∩  A3  ∩ ...  ∩ An =  ∩ Ai , i = 1 , 2 , 3 , ... , n 
and the other extension principle ( union ) 
A1  ∪ A2  ∪ ...  ∪ An =  ∪ Ai , i = 1 , 2 , 3 , ... , n 
About  DeMorgan Law , please see this .  
Basically, multi-sets' relations of union and intersection with complement , they can be represent as :