What Is a Singleton Set Definition Properties and Solved Examples
Understanding a singleton set is important for exams and practical maths. Whether you’re answering sets questions, classifying types, or preparing for competitive tests, knowing this basic concept lets you solve problems about grouping, subsets, and set notation with confidence.
Formula Used in Singleton Set
The standard formula is: \( n(A) = 1 \), where \( n(A) \) is the number of elements in singleton set \(A\).
Here’s a helpful table to understand singleton set more clearly:
Singleton Set Table
| Word | Value | Is This a Singleton Set? |
|---|---|---|
| {12} | 12 | Yes |
| {15, 18} | 15, 18 | No |
| {‘a’} | a | Yes |
| { } | — | No (Empty Set) |
This table shows how the pattern of singleton set appears regularly in real cases. Each singleton set has exactly one distinct element—never more, never less.
Worked Example – Solving a Problem
1. Write the singleton sets which are subsets of A = {3, 6, 9}.Step 1: List the elements of A: 3, 6, and 9.
Step 2: A singleton subset contains only one element. So, list each element by itself in curly brackets:
Step 3: Each of these sets has exactly one element, so each is a singleton set.
Final Answer: The singleton sets which are subsets of A are {3}, {6}, and {9}.
2. Find the powerset of the singleton set {m}.
Step 1: The set is A = {m}.
Step 2: List all subsets:
Step 3: Form the powerset by collecting these:
Final Answer: The powerset of {m} is {{}, {m}}.
Learn more about concepts like powerset and subsets to see how singleton sets fit into larger topics.
Practice Problems
- List all singleton sets that are subsets of B = {7, 8, 9, 10}.
- Which of these are singleton sets: {2}, { }, {4, 5}, {{1}}?
- Write the singleton set that contains the letter ‘z’.
- How many singleton subsets does the set {a, b, c, d, e} have?
Common Mistakes to Avoid
- Confusing singleton set with an empty set or with a set that has more than one element.
- Forgetting that each element creates its own singleton subset.
- Writing repeated elements in a set—remember: sets list each item only once.
Real-World Applications
The concept of singleton set appears in computer programming, groups with only one member, and database filtering (selecting exactly one result). At Vedantu, you can connect singleton set theory to problems in logic, coding, and mathematical modeling for real-life use.
Singleton Set vs Empty Set
A singleton set has one element (like {7}), while an empty set has none ({}). Although the symbols look similar, their meanings—and how you solve related questions—are different. For deeper understanding, see Sets and Types of Sets.
Properties of Singleton Set
1. A singleton set contains only one element.
2. The cardinality (number of elements) is always one.
3. Every singleton set has exactly two subsets: itself and the empty set.
4. Singleton sets are always finite.
5. Each element of a set forms a unique singleton subset.
Learn more about properties of sets to explore other types and features.
Page Summary
We explored the idea of singleton set, example problems, and why single-element sets are important for set theory and exams. Practice with more questions on Vedantu for a strong foundation in sets and related mathematical ideas.
FAQs on Singleton Set in Set Theory Explained Clearly
1. What is a singleton set in mathematics?
A singleton set is a set that contains exactly one element. In set theory, it is written using curly braces with a single value inside.
- Example: If A = {5}, then A is a singleton set.
- The number of elements in a singleton set is called its cardinality, which is 1.
- It is also known as a unit set.
2. How do you identify a singleton set?
A set is a singleton set if it contains only one distinct element. To identify it:
- Count the number of elements inside the braces.
- If the count (cardinality) is 1, it is a singleton set.
- Example: {0} and {a} are singleton sets, but {1, 2} is not.
3. What is the cardinality of a singleton set?
The cardinality of a singleton set is always 1. Cardinality means the total number of elements in a set.
- If A = {7}, then n(A) = 1.
- If B = {x}, then n(B) = 1.
4. What is the difference between a singleton set and an empty set?
The main difference is that a singleton set has one element, while an empty set has no elements.
- Singleton set example: {3}
- Empty set example: ∅ or { }
- Cardinality of singleton set = 1
- Cardinality of empty set = 0
5. Can you give an example of a singleton set?
An example of a singleton set is {10}. Here:
- The set contains only one element: 10.
- Its cardinality is 1.
- It can also be written in set-builder form, such as {x | x = 10}.
6. Is {0} a singleton set?
Yes, {0} is a singleton set because it contains exactly one element, which is 0.
- The number 0 is a valid element.
- The set has cardinality 1.
- Do not confuse {0} with the empty set ∅.
7. Is {∅} a singleton set?
Yes, {∅} is a singleton set because it contains exactly one element, which is the empty set.
- ∅ has no elements.
- {∅} has one element (the empty set itself).
- Therefore, its cardinality is 1.
8. What are the properties of a singleton set?
A singleton set has specific properties in set theory.
- It contains exactly one distinct element.
- Its cardinality is 1.
- It has exactly two subsets: the empty set ∅ and the set itself.
- It can be written in roster or set-builder form.
9. How many subsets does a singleton set have?
A singleton set has exactly 2 subsets. The number of subsets of a set with n elements is given by the formula 2n.
- For a singleton set, n = 1.
- Number of subsets = 21 = 2.
- Example: If A = {5}, subsets are ∅ and {5}.
10. What is the power set of a singleton set?
The power set of a singleton set contains exactly 2 subsets: the empty set and the set itself. If A = {a}, then:
- P(A) = {∅, {a}}
- The number of elements in the power set is 2.
- This follows the formula 2n where n = 1.
