Types of Sets

Empty Set or Null Set or Void Set
A set which does not contain any element is called empty set.
A null set is denoted by Φ or { }

Consider set A={x: x is a man living on the moon}
We know that there is no man living on the moon. Therefore, set A contains no elements and hence A is an empty set.

Let D={x: $\displaystyle \small x^{2}=4$, x is odd number}
Then D is empty set because the equation $\displaystyle \small x^{2}=4$ is not satisfied by any odd values of x.
 
The set {0} is not a null set, because this set contains one element '0'.
The set {Ï•} is not a null set, because this set contains one element Greek symbol 'Ï•'.

Singleton Set
A set consisting of single element is called singleton set.

Consider set P={x: x+6=0, x∈Z}={-6}
This is a singleton set since it contains only one element.

Finite Set
A set which consists of a definite/limited/countable number of elements is called finite set.

The set S={3,6,9,12} is a finite set because it consists of 4 elements.
The set of all days of a week is a finite set because it consists of 7 elements.

Cardinal number or Order of a finite set
The number of distinct elements of a finite set is called Cardinal number and is denoted by n(S).

Let D is a set of all letters in the word SCHOOL
D={S,C,H,O,L}
n(D)=5

Let F={x: x is a prime factor of 60}
F={2,3,5}
n(F)=3

An empty set which does not contain any element is also a finite set.

Infinite Set
A set which contains unlimited number of elements is called an infinite set.

Set of even natural numbers, {2,4,6...} is an infinite set.
{x: x∈N and x is prime}={2,3,5,7,11,13...} is an infinite set.

Equivalent Set
Two finite sets A and B are equivalent, if their cardinal numbers are same. i.e. n(A)=n(B)
Let E={a,b,c,d} and D={1,2,3,4}, then n(E)=4 and n(D)=4
E and D are equivalent sets.

Equal Set
Two sets A and B are said to be equal if they have exactly the same elements and denoted as A=B.
Otherwise the sets are said to be unequal and denoted by A≠B

Let A={1,4,8} and B={8,1,4}, then A=B because each element of A is in B and vice-versa.
Let C={x: x∈N, 2≤x≤6} and F={2,3,4,5,6}, then C=F.
Let G={x: x∈N, 10<x<11} and J={10.5}, then G≠J because 10.5∉G