# Types of Sets

What are the different types of sets?

The different types of sets are explained below with examples.

Empty Set or Null Set:

A set which does not contain any element is called an empty set, or the null set or the void set and it is denoted by ∅ and is read as phi. In roster form, ∅ is denoted by {}. An empty set is a finite set, since the number of elements in an empty set is finite, i.e., 0.

For example: (a) The set of whole numbers less than 0.

(b) Clearly there is no whole number less than 0.

Therefore, it is an empty set.

(c) N = {x : x ∈ N, 3 < x < 4}

Let A = {x : 2 < x < 3, x is a natural number}

Here A is an empty set because there is no natural number between
2 and 3.

Let B = {x : x is a composite number less than 4}.

Here B is an empty set because there is no composite number less than 4.

Note:

∅ ≠ {0} ∴ has no element.

{0} is a set which has one element 0.

The cardinal number of an empty set, i.e., n(∅) = 0

Singleton Set:

A set which contains only one element is called a singleton set.

For example:

A = {x : x is neither prime nor composite}

It is a singleton set containing one element, i.e., 1.

B = {x : x is a whole number, x < 1}

This set contains only one element 0 and is a singleton set.

Let A = {x : x ∈ N and x² = 4}

Here A is a singleton set because there is only one element 2 whose square is 4.

Let B = {x : x is a even prime number}

Here B is a singleton set because there is only one prime number which is even, i.e., 2.

Finite Set:

A set which contains a definite number of elements is called a finite set. Empty set is also called a finite set.

For example:

The set of all colors in the rainbow.

N = {x : x ∈ N, x < 7}

P = {2, 3, 5, 7, 11, 13, 17, ...... 97}

Infinite Set:

The set whose elements cannot be listed, i.e., set containing never-ending elements is called an infinite set.

For example:

Set of all points in a plane

A = {x : x ∈ N, x > 1}

Set of all prime numbers

B = {x : x ∈ W, x = 2n}

Note:

All infinite sets cannot be expressed in roster form.

For example:

The set of real numbers since the elements of this set do not follow any particular pattern.

Cardinal Number of a Set:

The number of distinct elements in a given set A is called the cardinal number of A. It is denoted by n(A).

For example:

A {x : x ∈ N, x < 5}

A = {1, 2, 3, 4}

Therefore, n(A) = 4

B = set of letters in the word ALGEBRA

B = {A, L, G, E, B, R}

Therefore, n(B) = 6

Equivalent Sets:

Two sets A and B are said to be equivalent if their cardinal number is same, i.e., n(A) = n(B). The symbol for denoting an equivalent set is ‘↔’.

For example:

A = {1, 2, 3} Here n(A) = 3

B = {p, q, r} Here n(B) = 3

Therefore, A ↔ B

Equal sets:

Two sets A and B are said to be equal if they contain the same elements. Every element of A is an element of B and every element of B is an element of A.

For example:

A = {p, q, r, s}

B = {p, s, r, q}

Therefore, A = B

The various types of sets and their definitions are explained above with the help of examples.

Set Theory

Sets

Objects Form a Set

Elements of a Set

Properties of Sets

Representation of a Set

Different Notations in Sets

Standard Sets of Numbers

Types of Sets

Pairs of Sets

Subset

Subsets of a Given Set

Operations on Sets

Union of Sets

Intersection of Sets

Difference of two Sets

Complement of a Set

Cardinal number of a set

Cardinal Properties of Sets

Venn Diagrams

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles 1. ### Rupees and Paise | Paise Coins | Rupee Coins | Rupee Notes

Dec 04, 23 02:14 PM

Money consists of rupees and paise; we require money to purchase things. 100 paise make one rupee. List of paise and rupees in the shape of coins and notes:

2. ### Months of the Year | List of 12 Months of the Year |Jan, Feb, Mar, Apr

Dec 04, 23 01:50 PM

There are 12 months in a year. The months are January, February, march, April, May, June, July, August, September, October, November and December. The year begins with the January month. December is t…