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:
- Cardinality is 1: Every singleton set contains exactly one element.
- Always Finite: Since the number of elements is one, it is a finite set.
- Subset Property: Every singleton set is a subset of another set.
- Equal Singleton Sets: Two singleton sets are equal if they have the same element.
- Power Set of a Singleton Set:
- 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:
- Roster Form: Listing the only element.
- Set Builder Form: Describing the property of the element.
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:x2=36,x>0} a singleton set?
- Solution: x2=36⟹x=±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
- Define a singleton set with one example.
- Write the cardinality of the set ( A = {7} ).
- Is the set ( B = {0} ) a singleton set? Justify your answer.
- Which of the following are singleton sets?
- (i) ( x:x2=9,x>0 )
- (ii) ( y:y2=25 )
- (iii) ( India )
- Find the singleton set from the following:
- (a) ( {1, 2, 3} )
- (b) ( {100} )
- (c) ( ∅ )
- If ( C = {x : x { is an even prime number} ), write ( C ) and state its cardinality.
- Is the set of vowels in the word "Sky" a singleton set? Explain.
- Represent the singleton set containing the solution of ( 2x + 3 = 7 ).
- Give one real-life example of a singleton set.
- Explain the difference between a singleton set and an empty set with examples.