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101 LearnersLast updated on September 29, 2025
Singleton Set
A singleton set is a set that contains exactly one element. It is also known as a unit set. In this lesson, we'll explore its definition, properties, and why it's useful in set theory.
What is a Singleton Set?
A singleton set has only one element. For example, set A = {1}. A singleton set has two subsets: the empty set and the set itself. If a set has n elements, then it has exactly n singleton subsets. For instance, set A = {a, e, i} has 3 elements, each element can form its singleton subset {a}, {e}, {i}.
An example of a singleton set is the set of even prime numbers: {2}, since 2 is the only even prime number.
Singleton Set Vs Empty Set
Singleton sets are different from empty sets, and their key distinctions are mentioned below.
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Singleton set |
Empty set |
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This set has exactly one element. |
This set has no elements. The total number of elements in this set is 0. |
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A singleton set is a subset of a set if its only element belongs to that set. |
It is always a subset of a set. |
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The cardinality of a singleton set is 1. |
The cardinality of an empty set is 0. |
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It is also called a unit set and is depicted as A = {a}. |
An empty set, also called a null set or void set, contains no elements and is written as {} or . |
Properties of the Singleton Set
Some important properties of a singleton set are listed below:
- A singleton set is a set with exactly one element.
- Its cardinality is always 1.
- It has two subsets, the null set and itself.
- The power set of a singleton set contains two subsets: the empty set and the singleton set itself. Therefore, its cardinality is 2.
- The empty set is a subset of every set, including all singleton sets.
- If two singleton sets have the same element, their union is also a singleton set.
Venn Diagram of a Singleton Set
A Venn diagram visually represents the relationship between sets. These relationships can include:
Intersection - elements that are shared.
Difference - elements that are unique to each set.
Union - all elements that are in either set or in both.
The Venn diagram given below represents a singleton set.
Is Zero Set {0} a Singleton Set?
A zero set is different from a null set. The zero set has exactly one element, that is zero, and its cardinality is 1. So, the zero set {0}, which contains only one element, is considered a singleton set.
Real-Life Applications of Singleton Set
A singleton set has many real-life applications. From technology to logic, singleton sets play an important role in many fields. Some of their applications are given below:
Error codes in software
When a program is running a specific error code, such as 404, it may be represented as a singleton set: {404}.
Storing a unique ID
A unique user ID is treated as a singleton set when it needs to be stored or accessed individually in a database.
Search results
When a database query has only one matching record, like a phone number, it is modeled as a singleton set.
Voting during elections
When a person casts a unique vote, let’s say candidate 1, their choice is expressed as {candidate 1}.
Mathematical definitions
Singleton sets are used in areas like topology and logic to define and test properties such as limits, continuity, or identity in mathematical models.
Common Mistakes and How to Avoid Them in Singleton Set
Even though singleton sets have just one element, they can be misunderstood, causing confusion and mistakes. Here are some common mistakes that students make; learning about them will help us avoid similar mistakes.
Mistake 1
Confusing singleton sets with null sets
Always remember that a singleton set will always have one element, and an empty set will never have any elements. Confusing the two leads to mistakes. For example, thinking {0} = is incorrect.
Mistake 2
Ignoring brackets while writing elements
A set must always be enclosed in curly brackets { }. For example, writing A = 7 looks like an equation, not a set. For it to be interpreted as a set, it should be written as A = {7}.
Mistake 3
Thinking that a singleton set must always contain 1
Students get confused that a singleton set is known as a unit set, so it must contain {1} as its element. Remember that a singleton set can have any single element like {0}, {-2}, {“name”}, etc.
Mistake 4
Confusing a set with one of its elements
{2} is a set containing the number 2, {{2}} is a set containing another set as its only element. These two notations are not the same.
Mistake 5
Treating a set as if it's just an element
A singleton set like {a} is not the same as the element a itself. Students often mistakenly write {a} = a, but that’s incorrect. The key difference is:
- a is just an element.
- {a} is a set that contains a as its only element.
Solved Examples of Singleton Set
Problem 1
What is the cardinality of set A = {8}
1
Explanation
The set has exactly one element, so its cardinality (number of elements) is 1. This makes it a singleton set.
Problem 2
Determine whether the set C = {x | x is a prime number less than 2} is a singleton set.
No, it is an empty set.
Explanation
There is no prime number less than 2. So the set is empty, not a singleton.
Problem 3
Is the set D = {“apple”} a singleton set?
Yes.
Explanation
A singleton set can have any single object, including numbers, words, or a name. In this case, it’s the name of a fruit. Since the set D contains only one element, it is a valid singleton set.
Problem 4
If E = {{5}}, is it a singleton set?
Yes
Explanation
The set E has exactly one element, that is, the set {5}. So it is a singleton set.
Problem 5
If F = {x | x2 = 25 and x > 0}, is F considered a singleton set?
Yes
Explanation
Upon solving the equation x2 = 25 x= 5, only x = 5 satisfies it. Since x > 0, F = {5} is the only possible element, making it a singleton set.
FAQs on Singleton Set
1. What is a singleton in mathematics?
2. How many elements are in a singleton set?
3.Are singleton sets finite?
4.Are singleton sets and unit sets the same?
5. What is an example of a singleton set?
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
