Difference Between Singleton Set and Empty Set Explained
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 What Is a Singleton Set? Definition, Examples, and FAQs for Students
1. What is a singleton set?
A singleton set is a set that contains exactly one unique element. In set theory, it is denoted as {a}, where 'a' is the sole element of the set. For example, {5} and {apple} are singleton sets.
2. What is a singleton information set?
A singleton information set in mathematics or game theory refers to an information set that contains only one possible outcome or state. This means the decision-maker knows exactly which situation they are in, reducing uncertainty to just a single case.
3. Is 1 a singleton set?
No, 1 itself is not a singleton set. However, the set {1} is a singleton set because it contains exactly one element. The number 1 is an element; {1} is a set containing the element 1.
4. What is the difference between empty set and singleton set?
The empty set (denoted by {} or ∅) contains no elements, while a singleton set has exactly one element. For example, {} is empty but {7} is a singleton. So, an empty set has cardinality 0, whereas a singleton set has cardinality 1.
5. What is the symbol of a singleton set?
A singleton set is represented using curly brackets with a single element inside. The general symbol is {a}, where 'a' is that single unique element of the set.
6. Give 5 examples of singleton sets.
Here are five examples of singleton sets:
• {2}
• {red}
• {Sunday}
• {π}
• {"A"}
7. What is the meaning of singleton set in Hindi?
सिंगलटन सेट (Singleton Set) वह समुच्चय है जिसमें सिर्फ एक ही सदस्य या तत्व होता है। उदाहरण के लिए, {5} एक सिंगलटन सेट है क्योंकि उसमें सिर्फ एक सदस्य है।
8. What are singleton sets in Java?
In Java programming, a singleton set refers to a set that contains exactly one element. The class Collections.singleton(element) is used to create such unmodifiable singleton sets.
9. Is a singleton set open or closed?
In mathematics and topology, a singleton set is always closed in most common topologies like the real numbers with standard topology. Whether it is open depends on the specific topology defined, but in common cases, singleton sets are closed sets.
10. Give 10 examples of singleton sets.
Examples of singleton sets include:
• {0}
• {7}
• {banana}
• {Monday}
• {π}
• {blue}
• {x}
• {25}
• {"one"}
• {3.14}
11. What is the definition of a singleton set?
A singleton set is a set which contains exactly one element. If S is a set and S = {a}, then S is called a singleton set.
12. Can you give a real-life example of a singleton set?
Yes, a real-life example of a singleton set would be the set of countries with the Taj Mahal, which is {India}, since only India has the Taj Mahal.
