A singleton set is a set that contains only one element. It is thus also called a unit set.
For example, a set of even prime numbers is a singleton set since only one prime number is even = {2}; all others are odd.
Here are a few more examples of singleton sets:
B = {12}
D = {x | 17 < x < 19 and x ∈ ℤ}
Symbol
If ‘A’ is a singleton set containing the element ‘a,’ it is written as:
A = {a}
Properties
Cardinality
Since there is always only one element in a singleton set, its cardinality is 1
Thus, if A is a singleton set, its cardinality is |A| = n(A) = 1.
Closed
Since the complement of any singleton set is an open set, we conclude that the singleton sets are always closed.
Finite Set
Since a singleton set has exactly one element, it is a finite set.
Subset
Empty sets are subsets of every set, as well as of the singleton sets. Thus, the number of subsets of a singleton set is two: the empty set (ɸ) and the set itself.
Also, since singleton sets have only one element, they are subsets of all sets containing that element.
Power Set
The power set of a set consists of all subsets of the given set. Thus, the power set of any singleton set always contains 2 elements.
If A = {a} is a singleton set, its power set is P(A) = {ɸ, {a}}
Thus, any singleton set has a powerset with a cardinality of 2.
For example,
If B = {12}, the power set of B is P(B) = {ɸ, {12}} and its cardinality |P(B)| = 2
Singleton Set Venn Diagram
The Venn diagram of a singleton set is a circle with only one element, as shown.
Singleton Set vs. Empty Set
Singleton Set
Empty Set
Contains only one element.
Does not contain any elements
The singleton set A is represented by A = {a}
An empty set is represented by ɸ = { }
Its cardinality is 1, means n(A) or |A| = 1
Its cardinality is 0, means n(ɸ) = |ɸ| = 0
Also known as a unit set.
Also known as a null or void set.
It has two subsets: the empty set and the set itself.
It has only one subset: the empty set.
Is the Zero Set {0} a Singleton Set
The zero set {0} is a set with ‘0’ as its only element. Thus, it is a singleton set.
However, the zero set {0} is not an empty set.
Solved Examples
Define the set of vowels in the color BROWN. Is it a singleton set?
Solution:
Let X be a set containing vowels in the color BROWN Then, X = {O} has 1 element. Thus, it is a singleton set.
Find the cardinality of the power set of X = {x | 78 < x < 80 and x is an integer}
Solution:
Given, X = {x | 78 < x < 80 and x is an integer} = {79} Now, its power set is {ɸ, {79}} Thus, the cardinality of the given power set is 2
Which of the following is a singleton set? a) A = {the prime numbers less than 10} b) B = {x | x2 = 16 and x is a natural number}
Solution:
a) The prime numbers less than 10 are 2, 3, 5, and 7 Here, A = {2, 3, 5, 7} has 4 elements. Thus, A is not a singleton set. b) x2 = 16 ⇒ x = ± 4 Since x is a natural number, B = {4} Thus, B is a singleton set.
