Sets and Their Representations

Set is a collection of well defined objects. By well defined objects, we mean the definition should not vary it  should be definite.
Example 1- Collection of vowels of English alphabet. The collection will be a, e, i, o, u. This is a well defined collection.
Example 2- Collection of good maths books available in market. Here, the collection will vary form person to person because of different personal choices and preferences.
There are two methods of representing a set : 
(i) Roster or tabular form
(ii) Set-builder form.
Example- Collection of vowels of English alphabet.
A= {a, e, i, o, u} this is the roster form of a set.
Here, A is represented as a set, and the elements of set are enclosed in { }.
Set builder form, A= {x: x is a vowel in English alphabet}
x represents the elements in the enclosed brackets { }.
Note:
1) Order is immaterial. That the sequence in which the elements appear is not important, even if the sequence is changed the meaning remains  same.
Take the same example- Collection of vowels of English alphabet.
A= {e, i, o, u, a}
2) Same elements are not repeated.
Example- B= set of all letters of the word SCHOOL
Roster form: B= {S, C, H, O, L}
Set-builder form: B= {x:x is letter used in word SCHOOL}
3) C= Set of all natural numbers.
Set-builder form C = {x :x ∈ N}
Roster form C= {1, 2, 3, 4, ...}
As natural numbers are infinite so we will represent them with three dots.
∈ means belongs to and ∉ means does not belongs to.
Example- A= {1, 2, 3, 4, 5, 6}
Here, 2 ∈ A, 9 ∉ A, 8 ∉ A, 5 ∈ A
We give below a few more examples of sets used particularly in mathematics, viz. 
`"N"` : the set of all natural numbers
`"Z"` : the set of all integers 
`"Q"` : the set of all rational numbers 
`"R"` : the set of real numbers 
`"Z"^+` : the set of positive integers 
`"Q"^+` : the set of positive rational numbers, and 
`"R"^+` : the set of positive real numbers.

1) Roster or Tabular method or List method
2) Set-Builder or Rule Method
3) Venn Diagram