Introduction– The concept of sets is widely used in mathematics as well as in other fields such as engineering, computer sciences etc.
The whole structure of Pure or Abstract Mathematics is based on the concept of sets.
German mathematician Georg Cantor (1845-1918) introduced and developed the theory of sets and subsequently many branches of modern Mathematics have been developed based on this theory.
Sets
A set is a collection of well-defined objects which are called the elements or members of the set.
Here by well defined means, given a particular collection of objects as a set and a particular object, it must be possible to determine whether that particular object is a member of the set or not.
We generally use capital letters to denote sets and lowercase letters for elements.
Suppose an element ‘a’ is from a set X, then we represent it as-
Which means the element ‘a’ belongs to the set X.
If the element is not form the group then we use ‘not belongs to’.
Representation of sets
Sets are represented in the following two methods :
1. Roster or tabular method
2. Set-builder or Rule method
Roster or tabular method – In this way, the elements of the set are separated by comma and contained in the bracket-{ }.
Example: A = {2, 3, 5, 7, 11, 13}
B = {1, 3, 5, 7, …}
Set-builder or Rule method
The second way is that we define the characteristic of the element.
For example-
Suppose we have a set ‘A’ of all odd integers which are greater than 3.
Then we can represent it as-
A = {x | x is an odd integer, x>3 }
Subsets
Let every element of set X is also an element of a set Y, then X is said to be the subset of Y.
Symbolically it is written as-
Which is read as- X is the subset of Y.
Note- If X = Y then
Proper sub-set
A set X is called proper subset the set Y is-
1. X is subset of Y
2. Y is not subset of X
Note- every set is a subset to itself.
Equal sets-
If X and Y are two sets such that every element of X is an element of Y and every element of Y is an element of X, the set and set Y will be equal always.
We write it as X = Y and read as X and Y are identical.
Super set-
If X is a subset of Y, then Y is called a super set of X.
Null set-
A set which does not contain any element is known as null set.
We denote a null set by ∅
The null set is a subset of every set.
Singleton-
A set which has only one element is called singleton.
Example- A = {10} and B = {∅}
Finite and infinite sets:
Definition: A set containing finite number of distinct elements so that the process of counting the elements comes to an end after a definite stage is called a finite set; otherwise, a set is called an infinite set.
Example: State which of the following sets are finite and which are infinite.
(i) the set of natural numbers N – infinite set
(ii) the set of concentric circles in a plane- infinite set
(iii) the set of rivers on the earth- finite set
Equal sets:
Definition : Two sets A and B are equal sets if every element of A is an element of B and every element of B is also an element of A. In other words, A is equal to B, denoted by A = B if A and B have exactly the same elements. If A and B are not equal, we write 
Power set:
Definition : The set consisting of all the subsets of a given set A as its elements, then we call it the power set of A and is denoted by P(A) . Thus,
