Singleton Set

Singleton set is a set with only one element. An example of a singleton set is an even prime number since there is only one prime number and all other primes are odd. Thus, the set of even prime numbers is a singleton set.

In this article, we will explore the singleton set, singleton set definition, singleton set symbol, singleton set examples, the cardinality of the singleton set, the power set of the singletons set, and others in detail.

What is Singleton Set?

A set is referred to as a singleton set when the number of elements in the set is one. The set of the form S = {a} is called a singleton set. The singleton sets are always finite as it has only 1 element.

Singleton Set Definition

Singleton set is defined as a set containing only one element. In other words, the set with one element is called a singleton set. The singleton set is also known as a unit set as the number of elements in the singleton set is a unit.

Singleton Set Symbol

The symbol of the singleton set is written as:

P = {x}

where:

• P is a singleton set with an element x

Singleton Set Examples

All the sets with only one element are singleton sets. Some examples of singleton sets are:

• Set of even prime numbers
• Set B = {10}
• Set Q = {|x|: x² = 16}

Cardinality of Singleton Set

The cardinality of singleton set is 1 as the number of elements present in the singleton set is 1. Since the singleton set contains only one element the cardinal number of singleton set is 1.

Cardinality of Singleton Set = 1

Power Set of a Singleton Set

The power set of a singleton set contains two elements the null set and the singleton set itself. The cardinality of the power set of a singleton set is 2 as it contains only two elements.

Properties of the Singleton Set

Some properties of singleton set are:

• Singleton set has only 1 element.
• Singleton set has the cardinality one.
• The singleton sets are always finite sets as the number of elements in the singleton set is 1.
• The number of subsets of singleton set is 2 i.e., empty set and the singleton set itself.
• The cardinality of power set of singleton set is 2.
• Empty set is subset of every singleton set.
• The union of two singleton set is singleton when the two sets are same.

Venn Diagram of a Singleton Set

Venn diagrams are the pictorial representations of any set. Below Venn diagram represents a singleton set.

Venn Diagram of a Singleton Set

Singleton Set Vs Empty Set

The table below gives us the difference between the singleton set and empty set.

Singleton Set	Empty Set
A set with only 1 element is called singleton set.	A set with no element is called as empty set.
The cardinality of singleton set is one.	The cardinality of empty set is zero.
Singleton set is also called as unit set.	Empty set is also called as null set or void set.
Singleton set has two subsets: Empty Set and the Singleton Set itself.	Empty set has only one subset i.e., Empty Set itself.

Is Zero Set {0} a Singleton Set?

Zero set is a set with one element zero. The cardinality of zero set is 1. So, the zero set i.e., {0} is a singleton set. Zero set and null set are different as zero set represents a set with one element zero whereas the null set is a set with no elements. Since a set with one element is singleton set so, the zero set is also a singleton set.

Zero set S = {0} is a singleton set

Solved Singleton Set Examples

Example 1: Check whether the given sets are singleton sets or not?

(i) Set of natural number less than 5.

(ii) S = {52}

(iii) Set of all the vowels in RAT.

Solution:

(i) Set of natural number less than 5.

Set of natural number less than 5 = {1, 2, 3, 4}

Since the above set contains four elements so, it is not a singleton set.

(ii) S = {52}

Since S contains only one element 52 so it is a singleton set.

(iii) Set of all the vowels in RAT.

Set of all vowels in RAT = {A} which contains only one element A so, it is a singleton set.

Example 2: Find the power set of the singleton set A = {{4}}.

Solution:

The power set of the singleton set A = {𝜙, {{4}}}

Example 3: Find the singleton subsets of the set P = {x, y, z}

Solution:

The singleton subsets of the set P = {x, y, z} are {x}, {y} and {z}.

Example 4: What is the number of singleton subsets in the set X = {2, 5, 10, 26, 52}

Solution:

The number of singleton subsets in the set X = {2, 5, 10, 26, 52} is 5 as the set X contains 5 different elements which individually forms 5 singleton sets.

Practice Problems on Singleton Set

Q1. Write all the singleton subsets of the set {1, 6, 9, 15}.

Q2. What is the power set of given singleton set S = {p}.

Q3. Check whether the set P = {x: x³= 8} is singleton set or not?

Singleton Set FAQs

Define Singleton Set with Example?

The singleton set is defined as the set with only one element. An example of singleton set is A = {7}.

Is 𝜙 a Singleton Set?

No, 𝜙 is not a singleton set as the 𝜙 represents null set i.e., the set with no element.

What is the Number of Elements in a Singleton Set?

The number of elements in a singleton set is one.

What is the Cardinal Number of Singleton Set?

The cardinal number of the singleton set is one as it contains only one element.

Is a Singleton Set Finite or Infinite?

The singleton sets are always finite sets as it contains only one element which is finite.