Singleton set

Sets are a foundational concept in mathematics, central to various fields such as statistics, geometry, and algebra. Singleton sets are the basic elemental mathematical concepts that one could use to explain real-life examples of the said topic. Consider a typical classroom scenario: suppose only a teacher who intends to classify learners based on their preferences in class contents would need such a relationship. Through this, she names a set of all the students who have an interest in mathematics to ensure she only tests the skills and knowledge of those who have an interest in the subject.

This article will cover the concept of the Singleton Sets. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more.

Singleton SETS

Definition of Set: A set is simply a collection of distinct objects, considered as a whole. These objects, called elements or members of the set, can be anything: numbers, people, letters, etc. Sets are instrumental in defining and working with groups of objects that share common properties.

It is a well-defined collection of distinct objects and it is usually denoted by capital letters A, B, C, S, U, V…...

If a is an element of a set A, we say that “ a belongs to A” The Greek symbol ∈ (epsilon) is used to denote the phrase ‘belongs to’.

Singleton set: A set which is having only one element is called a singleton set.

Example: {3}, {b},

{{1,2,3}} is also singleton as it has one element which is a set

{φ} is also a singleton set

Note: Empty and singleton sets are finite sets

Finite Sets:

A set which is empty or consists of a finite number of elements is called a finite set.

Example: A set of natural numbers up to 10.

A = {1,2,3,4,5,6,7,8,9,10}

Properties Of Singleton Set

• A singleton set contains exactly one element, its cardinality is 1
• The singleton set has two subsets.
• Each element in a singleton set is unique within that set.
• The two subsets of a singleton set are the null set, and the singleton set itself.
• The power set of a singleton set includes the empty set and the singleton set itself.

Summary: A singleton set includes only one set. They are used to distinguish between the element itself and a set containing that element. The main application of sets and their different form is in various fields such as statistics, geometry, and algebra.

Solved Examples Based On the Singleton Sets:

Example 1: Which of the following is not a singleton set?

1) Set of all natural numbers which are neither a prime nor a composite number.

2) Set of even prime numbers.

3) Set of numbers that divide 12 and 20.

4) S={}.

Solution:

(1) set = {1}

(2) set = {2}

(3) There are two numbers that divide 12 and 20 i.e. 2 and 4. So, the set is {2, 4}.

(4) {} is a singleton set with one element .

Hence, the answer is the option (3).

Example 2:

Which of the following is a singleton set?

1)

2)

3)

4) None of these

Solution

(1)

Number of elements is zero. So, not a singleton set.

(2)

this set has one element:

So, it is a singleton

(3) This set is .

So, not a singleton.

Hence, the answer is the option (2)

Example 3: Which of the following is a singleton set

1)

2)

3)

4) None of these

Solution

(a) A is singleton as it contains only one number. If number of students is 20 , then

(b)


As can never be negative
is always equal or greater than 2. Hence all real value of x satisfy Hence B is an infinite set.

(c) As there are infinite real numbers between 1 and 3 , So c is an infinite set .

Note: If the set included only natural numbers , then it was a singleton set: is a singleton set

Example 4: Which of the following is not a singleton set?

1) { x: x>5 and x<7 and x is a natural number}

2)

3)

4)

Solution

Singleton Set: A set that has only one element. eg. {3}, {b}

In this Question,

A= {6},

B={3},

C={-3}

D)

So, this set has 2 elements, and thus it is not a singleton

Hence, the answer is the option (4).

Example 5: Which of the following is not a singleton set?

1) Set of all natural numbers which are neither a prime nor a composite number.

2) Set of even prime numbers.

3) Set of numbers that divide 12 and 20.

4) S={}.

Solution

(1) set = {1}

(2) set = {2}

(3) There are two numbers that divide 12 and 20 i.e. 2 and 4. So, the set is {2, 4}.

(4) {} is a singleton set with one element .

Hence, the answer is the option (3).