Singleton Set: Definition, Properties, Cardinality & Applications  

1.0What is a Singleton Set?

A singleton set, also known as a unit set, is a set that contains exactly one element. This single element can be anything—a number, a variable, an object, or even another set. The defining characteristic of a singleton set is its cardinality, which is always 1. The term "cardinality" refers to the number of elements in a set.

Think of it like a small bag that can only hold one item, no matter what that item is. For example, a bag with just one blue ball is a singleton set, and so is a bag with just one red ball.

A set A is a singleton set if and only if its cardinality, denoted as ∣A∣ or n(A), is equal to 1. Mathematically, we can express a singleton set as: 

A={x}

where x is the single element contained within the set.

Examples of Singleton Sets:

• A={5} is a singleton set. The number 5 is its single element.
• B={−10} is a singleton set.
• C={The current Prime Minister of India} is a singleton set.
• D={{a,b,c}} is a singleton set. Its single element is the set {a,b,c}.
• E={∅} is a singleton set. Its single element is the empty set. This is a very important and often misunderstood example.

Note: A singleton set is always finite since it has only one element.

2.0Properties of Singleton Set

Some important properties of singleton sets include:

1. Cardinality is 1: Every singleton set contains exactly one element.
2. Always Finite: Since the number of elements is one, it is a finite set.
3. Subset Property: Every singleton set is a subset of another set.
4. • Example: {2}⊆{1,2,3,4}.
4. Equal Singleton Sets: Two singleton sets are equal if they have the same element.
5. • Example: {5}={5}.
5. Power Set of a Singleton Set:
6. • If A={x}, then P(A)={ϕ,{x}}.
   • Thus, the power set of a singleton set has 2 elements.

3.0Cardinality of Singleton Set

The cardinality of a set is the number of elements in the set.

• For a singleton set, the cardinality is always 1.
• Example: A={10}, then n(A)=1.

This property is a direct and simple way to identify singleton sets.

4.0Representation of Singleton Set

Singleton sets can be represented in two ways:

1. Roster Form: Listing the only element.
2. • Example: {a}.
2. Set Builder Form: Describing the property of the element.
3. • $\{x:x^2=25,x>0\}=\{5\}.$

5.0Difference Between Singleton Set and Other Sets

Key Difference: A singleton set is different from a null set. The null set has 0 elements, while the singleton set has exactly 1 element.

6.0Importance of Singleton Set 

• Forms the basis of understanding relations and functions.
• Important for questions related to cardinality and subsets.
• Appears in NCERT exercises which often frame JEE problems.
• Helps in understanding power sets, union, and intersection at a deeper level.

7.0Applications of Singleton Set in Real Life and Mathematics

Mathematics Applications:

• In functions, the image of a constant function is a singleton set.
• • Example: If f(x)=5 for all x, then range of f = {5}.
• In probability, events like “getting a 6 when a die is thrown” form singleton sets.
• In geometry, the intersection of two coinciding lines at a single point results in a singleton set.

Real-Life Applications:

• A classroom with only one student present → {student}.
• A basket with only one fruit → {apple}.
• A country’s national capital (only one city is the capital) → e.g., {New Delhi}.

8.0Solved Examples on Singleton Set

Example 1:
Is $A=\{x:x^2=36,x>0\}$ a singleton set?

• Solution: $x^2=36⟹x=\pm 6$ Since x>0, only x=6.
  Thus, A={6}, which is a singleton set.

Example 2:
Find the power set of A={π}.

• Solution: P(A)={ϕ,{π}}.
  It has 2 elements.

Example 3:
State whether B={x:x<1,x∈N} is a singleton set.

• Solution: Natural numbers start from 1, but no natural number is less than 1.
  Hence, B=ϕ → It is an empty set, not a singleton set.

9.0Practice Questions on Singleton Set

1. Define a singleton set with one example.
2. Write the cardinality of the set ( A = {7} ).
3. Is the set ( B = {0} ) a singleton set? Justify your answer.
4. Which of the following are singleton sets?
5. • (i) ( $x:x^2=9,x>0$ )
   • (ii) ( $y:y^2=25$ )
   • (iii) ( India )
5. Find the singleton set from the following:
6. • (a) ( {1, 2, 3} )
   • (b) ( {100} )
   • (c) ( $\varnothing$ )
6. If ( C = {x : x { is an even prime number} ), write ( C ) and state its cardinality.
7. Is the set of vowels in the word "Sky" a singleton set? Explain.
8. Represent the singleton set containing the solution of ( 2x + 3 = 7 ).
9. Give one real-life example of a singleton set.
10. Explain the difference between a singleton set and an empty set with examples.