sets

                                            Sets

To describe the definition of set let us concentrate on the given example, suppose  from market you purchased   bread,candle, biscuit,etc all the objects you purchased are well defined and their collection is called as set what about the collection of top 10 good ballers of the world or collection of best 5 comedian of the world, does they make set ? Answer is no, not at all because the terms good footballer and best comedian are vague and they are not well defined. From the above short example we understand the real definition of set that is
Definition:- Well defined collection of objects is called set , that means for a collection to be a set it is necessary that it should be  well defined
Some examples of set
1-collection of all English alphabets is a set
2-collection of all real numbers  is a set
3-collection of all boys in school is a set
4-collection of all past Presidents of USA is a set
5-collection of all countries in Arab world is a set
Before we learn about the representation of set in different forms
Let us first know about some standard notation to represent set
            N: The set of natural numbers
            W: The set of whole numbers
            Z: The set of integers
            Z- : The set of negative integers
            Z+:The set of positive integers
            Q: The set of rational Numbers
            R: The set of real numbers
            C :The set of all complex numbers
Some frequently used symbols are also  given as
∈ :belong to
∉  : 'does not belong to'
 ∃ :  There exists at least

              REPRESENTATION OF SETS    
      Set can be represented in two forms as per our convenience
1:- roaster form  2:- set builder form
Roaster form> In roaster form the set is expressed by listing of all elements in a curly brackets  which are separated  by commas
For an example  collection of all vowel  in an English alphabet
Which is  "'a,e,i,o,u' " it can be represent in roaster form as
                      A= {a,e,i,o,u}
In above example every element of set "A"is expressed clearly with commas in between them. Now let us look at some illustrations to get more idea regarding this method
Set of natural numbers N
N= {1,2,3,4,5,6,....so on}
Set of all whole numbers
W= {0,1,2,3,4,5,6...….so on}
Set of all integers
I or Z = {so on .....,-6,-5,-4,-3,-2,1,0,1,2,3,4,5,6,......so on}.In above illustrations the sets  are expressed in a roaster form. One fact should  be kept in mind that in a roaster form elements are  not repeated  in a set for an example the collection of alphabet in a word MATHEMATICS  is as
D= { m,a,t,h,e,i,c,s}
 builder form> In set builder form method the set is expressed in terms of its  general property and that general property gives the complete description of a  given set.
For an example let "A"be set of  all even natural numbers  then it can be represented in set builder form as
       A= { x; x is a natural number and x= 2n for n∈N}
Or,   A= { x;x∈N,x=2n,n∈N}
This is how we represent the set in roaster form method. Some  more examples are given below
😌Set of all real numbers greater then -1 & less than 1 can be described as { x∈R where -1< x< 2}
😌Set of all alphabets in a word 'PROBABILITY' can be represented in set builder form as { x:x is a letters in a word 'probability'}
😌Set of reciprocals of all natural numbers can be represented in set builder form as {x:x=1/n where n∈N}
😌Set of odd natural numbers less than 10 can be represented in set builder form as {x:x=2n–1,n∈N, x<10}
While learning mathematics it is important to grip your concepts by practice which is essential for learning mathematics so do some examples on your ownown.  

                                    CLASSIFICATION OF SET
 Sets are mainly classified into two groups or we can say that they are of two types 1:-finite set          2:-infinite set
1:-Finite set:-A set whose elements  are  countable to some limit is called as finite set. Finite set  are of two types
 A)empty set :–set which does not contain any element is called empty set. It is also known as null set or void set and it is denoted  by Ø(read as fie)or { }.
Fact regarding null set:- Null set has no element so the condition of becoming a subset is automatically satisfied. Therefore null set is a subset of every set. Subset of a set will be explained in another section.
B)Non empty set:- set which contains one or more than one element is called non empty set .The set which contains single element is called by some special name  as Singleton set. Now all the above discussion will be clear from the given example let us consider a classroom in which there are 60 students 40 of them study English and 20 Hindi. Let A be set of students who just study English and B be a set of students who just study Hindi then they can be represented in set builder form as
    A= {x:x is the no of students who just study English}
And also B= { y:y is the no of students who just study Hindi}
 Give the number of students who study both English and Hindi
Clearly none of any student study both English and Hindi  but this statement can be mathematically defined by an empty set as let E be a set of students who study both English and Hindi then
      E={ Ø }or { }
Note:- some students are confused why we use empty set, as the basic concept of set is the collection of well defined objects  and there is no element in an empty set then how can it be regarded as set? The answer is simple if we look at the definition of a set the word well defined means the things or objects that we include in a set are noticeable . In above example it is well and clear that there is no student who study both English and Hindi so we can express the statement by marking it as empty set

    2:–Infinite set:-A set  whose elements are uncountable but can be expressed as a set is called as infinite set. For example
            a)Set of all natural numbers
            b)Set of all real numbers
            c)Set of all fractions
             d)set of all possible fractions in between 1and 2
          e)Set of all stars in the universe
               components of sets
Let  A,B,C,D,E,F,G, be the Singleton  or non Singleton sets  since, they are all well defined as a definition of set  therefore, they can be putted into a single set or we can say that they could be represented by a single set ( say Z)  and that single set Z is called as universal set for its element sets so we can write it as
  Z={ (A),(B),(C),(D),(E),(F),(G)} the elements in this set are called as its subsets. Thus, the subset is a set contained in another set as its element. Subset of any set can be of two types 1)proper subset  2)improper subset
1)proper subset:- Let 'A' be a set containing subset 'a' then 'a' is said to be the proper subset of a set 'A' if and only, when all element of a subset 'a' is contained in a set 'A' but not all elements of 'A' are contained in 'a'. It is denoted by ⊂(read as proper subset of). Some examples of proper subset are given at the end of the chapter
2)Improper subset:- A subset 'a' is said to be an improper subset of a set 'A' if and only if every element of a subset 'a' is contained in a set 'A' and all elements in a set 'A' are contained in a subset 'a'. It is denoted by ⊆(read as improper subset of)
  Some examples of improper subset are also given at end of the chapter
                              Venn diagram
 British mathematician John Venn (1834 −1883 AD) introduced the concept of diagrams to represent sets. According to him set is represented by the  diagrammatic system in which  universal set is represented by the interior of a rectangle and other sets or subsets are represented by interior of circles within the rectangle. Reason for the development of the idea was to make the problems in set theory more convenient and easier or we can say that to express complex mathematical situations in an easier way. We will use the the concept in some mathematical problems later on.let  U be a universal set and A,B,C,D be its subsets then, they can be represented by a Venn diagram as