Singleton Set in Maths: Definition, Examples, symbol

Q: What do you mean by Singleton Set?

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

Q: What Is The other name Of the Singleton Set?

The singleton set is also called a unit set.

Q: What is the cardinality of a singleton set?

The cardinality of the singleton set is 1.

Q: What is the cardinality of a singleton set?

The cardinality of a singleton set is always 1, as it contains exactly one element.

Q: What is the power of singleton set?

The power of the singleton set is 2.

Q: How many subsets does a singleton set have?

A singleton set has exactly two subsets: the empty set {} and the singleton set itself. This is because we can either include the single element or not.

Q: Can a singleton set be an infinite set?

No, a singleton set cannot be an infinite set. By definition, it contains exactly one element, making it a finite set with a cardinality of 1.

Q: How does a singleton set relate to the concept of uniqueness in mathematics?

Singleton sets are often used to represent unique objects or concepts in mathematics. They ensure that we're dealing with exactly one instance of something, which is crucial in many mathematical contexts.

Q: How is a singleton set different from a unit set in set theory?

There is no difference between a singleton set and a unit set in set theory. These terms are used interchangeably to refer to a set containing exactly one element.

Q: How does the concept of a singleton set relate to the notion of a point in topology?

In topology, singleton sets often correspond to individual points in a topological space. The properties of these singleton sets (such as whether they are open or closed) can provide important information about the topology of the space.

Singleton set

Edited By Komal Miglani | Updated on Jul 02, 2025 06:38 PM IST

Consider a garden with various flowers like rose, tulip, lily etc. Now consider a single rose flower. Here the single rose flower is the singleton set of the set of flowers in the garden. A set with a single element is called a singleton set.

This article will cover the concept of the Singleton Sets. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more.

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This Story also Contains

Sets
Singleton set Definition
Cardinality of Singleton Set
Power Set of a Singleton Set
Is the Zero Set $\{0\}$ a Singleton Set?
Facts about Singleton Set
Solved Examples Based On the Singleton Sets

Singleton set

Sets

A set is simply a collection of distinct objects, considered as a whole. These objects, called elements or members of the set, can be anything: numbers, people, letters, etc. Sets are instrumental in defining and working with groups of objects that share common properties.

It is a well-defined collection of distinct objects and it is usually denoted by capital letters $A, B, C, S, U, V$ ......

If $a$ is an element of a set $A$ , we say that " a belongs to $A$ " The Greek symbol $\in$ is used to denote the phrase 'belongs to'.

Singleton set Definition

A set having one element is called a singleton set.
Let's look into singleton set examples.

The examples of singleton set includes ${3}, {b}$ , ${φ}$ , ${{1, 2, 3}}$ .
${{1, 2, 3}}$ is also singleton as it has one element which is a set
${φ}$ is also a singleton set

Singleton Set Venn Diagram

Let $A = {a}$ . The venn diagram of singleton set $A$ is

Properties of Singleton Set

The properties of singleton set includes,

A singleton set contains exactly one element, its cardinality is $1$
The singleton set has two subsets.
Each element in a singleton set is unique within that set.
The two subsets of a singleton set are the null set, and the singleton set itself.
The power set of a singleton set includes the empty set and the singleton set itself.

Cardinality of Singleton Set

The number of elements in a set is called the cardinality of the set. Thus, the cardinality of a singleton set is $1$ .

Power Set of a Singleton Set

The power set of a set is the set of all subsets of the given set. The number of subsets of a singleton set is two. One subset is the empty set $(\emptyset)$ , and the other is the set itself. Thus, the power set of any singleton set always contains only $2$ elements.

Is the Zero Set ${0}$ a Singleton Set?

The zero set ${0}$ is a set with “ $0$ ” as the only element. Thus, it is a singleton set.

Note that the singleton set ${0}$ is not to be confused with an empty set. An empty set is a set that has no element. It is completely null or void.

Singleton Set vs Empty Set

Singleton Set	Empty Set
A singleton set contains only one element.	An empty set has no elements.
A singleton set with element $x$ is denoted by ${x}$ .	An empty set is denoted by the symbol $\emptyset$ . It is also expressed as ${}$ .
If A $= {a}$ , then $n (A) = 1$ .	$n (\emptyset) = 0$
Other names are unit set, one-point set.	Other names are null set, void set.
Empty set is always a subset of a singleton set.	Empty set is the only subset of itself.

Facts about Singleton Set

Singleton sets are also known as unit sets.
The cardinality of a singleton set is $1$ .
Singleton sets are always finite sets.
Singleton sets are commonly used in mathematics to represent unique elements.
Every set that contains the single element of a singleton set, is its superset. In other words, a singleton set is a subset of every set that contains its single element.

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Solved Examples Based On the Singleton Sets

Question 1. Which of the following is not a singleton set?

1) Set of all natural numbers which are neither a prime nor a composite number.

2) Set of even prime numbers.

3) Set of numbers that divide $12$ and $20$ .

4) $S = {ϕ}$ .

Solution
(1) set $= {1}$
(2) set $= {2}$
(3) There are two numbers that divide $12$ and $20$ i.e. $2$ and $4$ . So, the set is ${2, 4}$ .
(4) ${ϕ}$ is a singleton set with one element $ϕ$ .

Hence, the answer is the option (3).

Question 2. Which of the following is a singleton set?
1) $ϕ$
2) ${ϕ}$
3) ${x : x^{2} - 4 = 0}$
4) None of these

Solution

(1) $ϕ$
Number of elements is zero. So, not a singleton set.
(2) ${ϕ}$
this set has one element: $ϕ$
So, it is a singleton
(3) This set is ${2, - 2}$ .

So, not a singleton.
Hence, the answer is the option (2)

Question 3. Which of the following is a singleton set
1) $A = {x : x$ is the number of students in a class $}$
2) $B = {x : x^{2} + 2 x + 3 ⩾ 0}$
3) $C = {x : 1 < x < 3, x \in R}$
4) None of these

Solution

(a) $A$ is singleton as it contains only one number. If number of students is $20$ , then $A = {20}$ .
(b)

$\begin{aligned} x^{2} + 2 x + 3 \\ = (x^{2} + 2 x + 1) + 2 \\ = (x + 1)^{2} + 2 \end{aligned}$

As $(x + 1)^{2}$ can never be negative
$∴ (x + 1)^{2} + 2$ is always equal or greater than 2 . Hence all real value of x satisfy $x^{2} + 2 x + 3 ⩾ 0$ . Hence $B$ is an infinite set.

Note: If the set included only natural numbers, then it was a singleton set: ${x : 1 < x < 3, x \in N}$ is a singleton set

Question 4. Which of the following is not a singleton set?
1) ${x : x > 5$ and $x < 7$ and $x$ is a natural number $}$
2) ${x : x^{2} = 9$ and $x$ is a positive integer $}$
3) ${x : x^{2} = 9$ and $x$ is a negative integer $}$
4) ${x : x^{2} - 3 x + 2 = 0$ and $x$ is a positive integer $}$

Solution

Singleton Set: A set that has only one element. eg. ${3}, {b}$

In this Question,

$A = {6}$ ,

$B = {3}$ ,

$C = {- 3}$

D) $x^{2} - 3 x + 2 = 0 \Rightarrow x = 1, 2$ and both are positive

So, this set has 2 elements, and thus it is not a singleton
Hence, the answer is the option (4).

Question 5. Which of the following is not a singleton set?

1) Set of all natural numbers which are neither a prime nor a composite number.
2) Set of even prime numbers.
3) Set of numbers that divide $12$ and $20$ .
4) $S = {ϕ}$ .

Solution
(1) set $= {1}$
(2) set $= {2}$
(3) There are two numbers that divide $12$ and $20$ i.e. $2$ and $4$ . So, the set is ${2, 4}$ .
(4) ${ϕ_{}}$ is a singleton set with one element $ϕ$ .

Hence, the answer is the option (3).

List of Topics Related to Singleton Set

Roster And Set Builder Form Of Sets	Intersection Of Set, Properties Of Intersection
Equal And Equivalent Sets	Difference Of Set
Universal Set	Complement Of A Set, Law Of Complement, Property Of Complement
Union Of Sets, Properties Of Union	De-morgan's Laws

Frequently Asked Questions (FAQs)

1. What do you mean by Singleton Set?

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

2. What Is The other name Of the Singleton Set?

The singleton set is also called a unit set.

3. What is the cardinality of a singleton set?

The cardinality of the singleton set is 1.

4. What is the cardinality of a singleton set?

The cardinality of a singleton set is always 1, as it contains exactly one element.

5. What is the power of singleton set?

The power of the singleton set is 2.

6. How many subsets does a singleton set have?

It has 2 subsets.

7. How many subsets does a singleton set have?

A singleton set has exactly two subsets: the empty set {} and the singleton set itself. This is because we can either include the single element or not.

8. Can a singleton set be an infinite set?

No, a singleton set cannot be an infinite set. By definition, it contains exactly one element, making it a finite set with a cardinality of 1.

9. How does a singleton set relate to the concept of uniqueness in mathematics?

Singleton sets are often used to represent unique objects or concepts in mathematics. They ensure that we're dealing with exactly one instance of something, which is crucial in many mathematical contexts.

10. How is a singleton set different from a unit set in set theory?

There is no difference between a singleton set and a unit set in set theory. These terms are used interchangeably to refer to a set containing exactly one element.

11. How does the concept of a singleton set relate to the notion of a point in topology?

In topology, singleton sets often correspond to individual points in a topological space. The properties of these singleton sets (such as whether they are open or closed) can provide important information about the topology of the space.

12. Can two different singleton sets be equal?

Two singleton sets are equal if and only if they contain the same element. For example, {a} = {a}, but {a} ≠ {b} if a and b are different.

13. Can a singleton set be a subset of itself?

Yes, every set, including a singleton set, is a subset of itself. This is because all elements of the set (in this case, the single element) are contained within itself.

14. What is the intersection of a singleton set with itself?

The intersection of a singleton set with itself is the singleton set itself. For example, {a} ∩ {a} = {a}, because the common element between the two sets is 'a'.

15. What is the union of a singleton set with the empty set?

The union of a singleton set with the empty set is the singleton set itself. For example, {a} ∪ {} = {a}, because the empty set doesn't add any new elements.

16. What is the Cartesian product of a singleton set with itself?

The Cartesian product of a singleton set {a} with itself is {(a,a)}. It results in a singleton set containing one ordered pair where both components are the single element 'a'.

17. What is the power set of a singleton set?

The power set of a singleton set {a} is {{}, {a}}. It contains all possible subsets of the singleton set, which are the empty set and the singleton set itself.

18. What is the result of taking the powerset of a powerset of a singleton set?

The powerset of a powerset of a singleton set {a} is {∅, {∅}, {{a}}, {∅, {a}}}. This is because the powerset of {a} is {∅, {a}}, and we then take the powerset of this resulting set.

19. Can a variable represent a singleton set?

Yes, a variable can represent a singleton set. For example, we might use S = {x} to define S as a singleton set containing the element x.

20. Can a singleton set be a proper subset of another set?

Yes, a singleton set can be a proper subset of another set. For example, {a} is a proper subset of {a, b, c}, as long as a, b, and c are distinct.

21. How does the concept of a singleton set relate to functions?

In function theory, a singleton set is often used to define the codomain of a constant function. For example, f(x) = c for all x in the domain can be thought of as a function whose codomain is the singleton set {c}.

22. What is the complement of a singleton set in a universal set?

The complement of a singleton set {a} in a universal set U is all elements in U except for 'a'. Mathematically, it's represented as U \ {a} or U - {a}.

23. How is a singleton set used in the definition of injective functions?

Singleton sets are used in the definition of injective (one-to-one) functions. A function f is injective if for every singleton set {y} in the codomain, its preimage f^(-1)({y}) is either empty or a singleton set.

24. What is the symmetric difference between a singleton set and itself?

The symmetric difference between a singleton set and itself is the empty set. For example, {a} △ {a} = {}, because there are no elements that are in one set but not the other.

25. How does the concept of a singleton set relate to the Axiom of Extensionality?

The Axiom of Extensionality states that two sets are equal if and only if they have the same elements. For singleton sets, this means {a} = {b} if and only if a = b, emphasizing the importance of the single element's identity.

26. What is a singleton set?

A singleton set is a set that contains exactly one element. It's the simplest non-empty set possible, representing a collection with a single member.

27. Can a singleton set be empty?

No, a singleton set cannot be empty. By definition, it must contain exactly one element. An empty set has zero elements, which is different from a singleton set.

28. How is a singleton set typically represented?

A singleton set is typically represented using curly braces {} with the single element inside. For example, {a} represents a singleton set containing only the element 'a'.

29. What's the difference between an element and its singleton set?

An element is a single object, while its singleton set is a collection containing only that object. For example, 'x' is an element, while {x} is the singleton set containing 'x'. They are not the same thing mathematically.

30. Can a singleton set contain another set?

Yes, a singleton set can contain another set as its single element. For example, {{1, 2, 3}} is a singleton set whose only element is the set {1, 2, 3}.

31. How are singleton sets used in the definition of a metric space?

In a metric space, the distance between a point and itself is always 0. This can be expressed using singleton sets: for any singleton set {x} in a metric space, the distance d(x,x) = 0.

32. What is the role of singleton sets in defining the concept of connectedness in topology?

In topology, a space is connected if it cannot be divided into two disjoint non-empty open sets. Singleton sets play a role in this definition, as they are often examples of connected spaces (in usual topologies) that cannot be further subdivided.

33. How is a singleton set used in the definition of a partition of a set?

In a partition of a set, singleton sets can be used as individual parts. For example, {{a}, {b}, {c}} is a partition of the set {a, b, c} where each part is a singleton set.

34. What is the relationship between singleton sets and the concept of an atom in set theory?

In set theory, an atom (or urelement) is an object that is not a set but can be an element of sets. Singleton sets are often used to "wrap" atoms, allowing them to be treated as sets while preserving their atomic nature.

35. Can a singleton set be both open and closed in a topological space?

Yes, in some topological spaces, a singleton set can be both open and closed (called a clopen set). This often occurs in discrete topological spaces where every subset, including singletons, is both open and closed.

36. How is a singleton set used in the definition of a Dirac measure in probability theory?

In probability theory, a Dirac measure (or Dirac delta measure) assigns a total probability of 1 to a singleton set and 0 to its complement. It's used to represent a discrete probability distribution concentrated at a single point.

37. What is the role of singleton sets in defining equivalence classes?

Singleton sets can represent equivalence classes in cases where each element is only equivalent to itself. In such cases, each equivalence class is a singleton set containing one element from the original set.

38. How does the concept of a singleton set relate to the idea of a universal set in set theory?

While a singleton set contains only one element, a universal set contains all elements under consideration. However, in some contexts, the universal set itself could be viewed as a singleton set in a larger framework, emphasizing the relative nature of these concepts.

39. Can a singleton set be a member of itself?

In standard set theory, a set cannot be a member of itself. Therefore, a singleton set {a} cannot be a member of itself unless we're working in a non-standard set theory that allows such constructions.

40. How is a singleton set used in the definition of a surjective function?

A function f is surjective (onto) if the image of the entire domain includes every element of the codomain. In terms of singleton sets, this means that for every singleton set {y} in the codomain, there exists an x in the domain such that f(x) = y.

41. What is the significance of singleton sets in the study of ordered pairs?

Singleton sets play a role in some definitions of ordered pairs. For instance, in the Kuratowski definition, the ordered pair (a,b) is defined as {{a}, {a,b}}, where {a} is a singleton set.

42. How does the concept of a singleton set relate to the idea of a minimal element in order theory?

In order theory, a singleton set often represents a minimal (or maximal) element in a partially ordered set. If {a} is a singleton subset of a partially ordered set, and no element is less than a, then a is a minimal element.

43. What is the relationship between singleton sets and the concept of a basis in linear algebra?

In linear algebra, singleton sets containing individual vectors can form a basis for a one-dimensional vector space. For example, {v} could be a basis for the one-dimensional subspace spanned by the vector v.

44. How are singleton sets used in the construction of natural numbers in set theory?

In set-theoretic constructions of natural numbers, such as the von Neumann ordinals, singleton sets play a role. For example, the number 1 is often represented as the singleton set {0}, where 0 is represented by the empty set.

45. What is the significance of singleton sets in the study of cardinal numbers?

All singleton sets have the same cardinality (size), regardless of what element they contain. This cardinality is denoted by the cardinal number 1, which is fundamental in the study of infinite cardinals and their arithmetic.

46. How does the concept of a singleton set relate to the idea of a constant sequence in analysis?

A constant sequence can be thought of as a function from the natural numbers to a singleton set. For example, the constant sequence (a, a, a, ...) corresponds to a function f: ℕ → {a}.

47. What is the role of singleton sets in defining simple graphs in graph theory?

In graph theory, singleton sets are often used to represent vertices. A simple graph with no edges and only one vertex would be represented as ({v}, ∅), where {v} is the singleton set containing the single vertex.

48. What is the significance of singleton sets in the study of fixed points in function theory?

A fixed point of a function f is a point x such that f(x) = x. In set notation, this can be expressed as {x} ⊆ f^(-1)({x}), where {x} is a singleton set. This formulation emphasizes the role of singleton sets in fixed point theory.

49. How does the concept of a singleton set relate to the idea of a degenerate case in mathematics?

Singleton sets often represent degenerate cases in various mathematical contexts. For example, a degenerate triangle (where all points are collinear) can be thought of as having a singleton set as its interior.

50. What is the role of singleton sets in defining the support of a function?

The support of a function is the set of points where the function is non-zero. For some functions, this support might be a singleton set, representing a function that is zero everywhere except at a single point.

51. How are singleton sets used in the definition of a Dirac delta function?

The Dirac delta function δ(x) is often described as a function that is zero everywhere except at x=0, where it is infinitely large. In a measure-theoretic sense, it can be thought of as assigning a measure of 1 to the singleton set {0} and 0 to all other singleton sets.

52. What is the significance of singleton sets in the study of isolated points in topology?

In topology, an isolated point of a set S is a point x such that the singleton set {x} is open in the subspace topology on S. This definition highlights the importance of singleton sets in characterizing the local structure of topological spaces.

53. How does the concept of a singleton set relate to the idea of a trivial group in group theory?

In group theory, a trivial group is a group with only one element. This can be represented as a singleton set {e}, where e is the identity element of the group. This emphasizes the connection between singleton sets and the simplest possible group structure.

54. How are singleton sets used in the definition of a probability mass function for discrete random variables?

For a discrete random variable X, the probability mass function p(x) gives the probability that X takes on the specific value x. This can be expressed in terms of singleton sets: p(x) = P(X ∈ {x}), where {x} is the singleton set containing x.

55. What is the significance of singleton sets in the study of well-ordered sets?

In a well-ordered set, every non-empty subset has a least element. Singleton sets play a special role in this context, as they are always well-ordered (trivially, since they have only one element), and they represent the simplest non-empty well-ordered sets.