Sets, in mathematics, are an organized collection of objects and can be represented in setbuilder form or in roster form.
Sets are represented as a collection of welldefined objects or elements and it does not change from person to person. A set is represented by a capital letter symbol and the number of elements in the finite set is represented as the cardinal number of a set.
The set theory defines the different types of sets symbols and operation performed.
In set theory, the operations of the sets are carried when two or more sets combined to form a single set under some of the given conditions. The basic operations on sets are:
Some of the most important set formulas are:
For any three sets A, B and C 
n ( A ∪ B ) = n(A) + n(B) – n ( A ∩ B) 
If A ∩ B = ∅, then n ( A ∪ B ) – n(A) + n(B) 
n( A – B) + n( A ∩ B ) – n(A) 
n( B – A) + n( A ∩ B ) – n(A) 
n( A – B) + n ( A ∩ B) + n( B – A) = n ( A ∪ B ) 
n ( A ∪ B ∪ C ) = n(A) + n(B) + n(C) – n ( A ∩ B) – n ( B ∩ C) – n ( C ∩ A) + n ( A ∩ B ∩ C) 
The sets are represented in curly braces, {}. For example, {2,3,4} or {a,b,c} or {Bat, Ball, Wickets}. The elements in the sets are depicted in either the Statement form, Roster Form or Set Builder Form.
In statement form, the welldefined descriptions of a member of a set are written and enclosed in the curly brackets.
For example, the set of even numbers less than 15.
In statement form, it can be written as {even numbers less than 15}.
In Roster form, all the elements of a set are listed.
For example, the set of natural numbers less than 5.
Natural Number = 1, 2, 3, 4, 5, 6, 7, 8,……….
Natural Number less than 5 = 1,2,3,4
Therefore the set is N = { 1, 2, 3, 4 }
The general form is, A = { x : property }
For example: Write the following sets in set builder form: A={2, 4, 6, 8}
Solution:
2 = 2 x 1
4 = 2 x 2
6 = 2 x 3
8 = 2 x 4
So, the set builder form is A = {x: x=2n, n ∈ N and 1 ≤ n ≤ 4}
Also, Venn Diagrams are the simple and best way for visualized representation of sets.
Let us take an example:
A = {1, 2, 3, 4, 5 }
Since a set is usually represented by the capital letter. Here A is the set and 1, 2, 3, 4, 5 are the elements of the set or a member of a set. The elements that are written in the set are in any order and it cannot be repeated. All the set elements are represented in small letter in case of alphabets. Also, we can write it as 1 ∈ A, 2 ∈ A etc. The cardinal number of the set is 5. Some commonly used sets are as follows:
The complement of any set, say P, is the set of all elements in the universal set that are not in set P. It is denoted by P’.
Properties of Complement sets:
Commutative Property :

Associative Property :

Distributive Property :

Demorgan’s Law :

Complement Law :

Idempotent Law And Law of null and universal set :
For any finite set A

Here are few sample examples, given to represent the elements of a set.
Example 1:
Write the given statement in three methods of representation of a set:
The set of all integers that lies between 1 and 5
Solution:
The methods of representations of sets are:
Statement Form: { I is the set of integers that lies between 1 and 5}
Roster Form: I = { 0,1, 2, 3,4 }
Setbuilder Form: I = { x: x ∈ I, 1 < x < 5 }
Example 2:
Find A U B and A n B and A – B.
If A = {a, b, c, d} and B = {c, d}.
Solution:
A = {a, b, c, d} and B = {c, d}
A U B = {a, b, c, d}
A n B = {c, d} and
A – B = {a, b}