Singleton set

Consider a garden with various flowers like rose, tulip, lily etc. Now consider a single rose flower. Here the single rose flower is the singleton set of the set of flowers in the garden. A set with a single element is called a singleton set.

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.

Sets

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 $\in$ is used to denote the phrase 'belongs to'.

Singleton set Definition

A set having one element is called a singleton set.

Let's look into singleton set examples.

The examples of singleton set includes $\{3\},\{b\}$, $\{\varphi\}$, $\{\{1,2,3\}\}$.
$\{\{1,2,3\}\}$ is also singleton as it has one element which is a set
$\{\varphi\}$ is also a singleton set

Singleton Set Venn Diagram

Let $A =\{a\}$. The venn diagram of singleton set $A$ is

Properties of Singleton Set

The properties of singleton set includes,

• 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.

Cardinality of Singleton Set

The number of elements in a set is called the cardinality of the set. Thus, the cardinality of a singleton set is $1$.

Power Set of a Singleton Set

The power set of a set is the set of all subsets of the given set. The number of subsets of a singleton set is two. One subset is the empty set $(∅)$, and the other is the set itself. Thus, the power set of any singleton set always contains only $2$ elements.

Is the Zero Set $\{0\}$ a Singleton Set?

The zero set $\{0\}$ is a set with “$0$” as the only element. Thus, it is a singleton set.

Note that the singleton set $\{0\}$ is not to be confused with an empty set. An empty set is a set that has no element. It is completely null or void.

Singleton Set vs Empty Set

Facts about Singleton Set

• Singleton sets are also known as unit sets.
• The cardinality of a singleton set is $1$.
• Singleton sets are always finite sets.
• Singleton sets are commonly used in mathematics to represent unique elements.
• Every set that contains the single element of a singleton set, is its superset. In other words, a singleton set is a subset of every set that contains its single element.

Solved Examples Based On the Singleton Sets

Question 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) $\mathrm{S}=\{\phi\}$.

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) $\{\phi\}$ is a singleton set with one element $\phi$.

Hence, the answer is the option (3).

Question 2. Which of the following is a singleton set?
1) $\phi$
2) $\{\phi\}$
3) $\left\{x: x^2-4=0\right\}$
4) None of these

Solution

(1) $\phi$
Number of elements is zero. So, not a singleton set.
(2) $\{\phi\}$
this set has one element: $\phi$
So, it is a singleton
(3) This set is $\{2,-2\}$.

So, not a singleton.
Hence, the answer is the option (2)

Question 3. Which of the following is a singleton set
1) $A=\{x: x$ is the number of students in a class $\}$
2) $B=\left\{x: x^2+2 x+3 \geqslant 0\right\}$
3) $\mathrm{C}=\{\mathrm{x}: 1<\mathrm{x}<3, \mathrm{x} \in R\}$
4) None of these

Solution

(a) $A$ is singleton as it contains only one number. If number of students is $20$, then $A=\{20\}$.
(b)

$
\begin{aligned}
& x^2+2 x+3 \\
& =\left(x^2+2 x+1\right)+2 \\
& =(x+1)^2+2
\end{aligned}
$

As $(\mathrm{x}+1)^2$ can never be negative
$\therefore(\mathrm{x}+1)^2+2$ is always equal or greater than 2 . Hence all real value of x satisfy $\mathrm{x}^2+2 \mathrm{x}+3 \geqslant 0$. 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: $\{\mathrm{x}: 1<\mathrm{x}<3, \mathrm{x} \in \mathrm{N}\}$ is a singleton set

Question 4. Which of the following is not a singleton set?
1) $\{x: x>5$ and $x<7$ and $x$ is a natural number $\}$
2) $\left\{x: x^2=9\right.$ and $x$ is a positive integer $\}$
3) $\left\{x: x^2=9\right.$ and $x$ is a negative integer $\}$
4) $\left\{x: x^2-3 x+2=0\right.$ and $x$ is a positive integer $\}$

Solution

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

In this Question,

$A= \{6\}$,

$B=\{3\}$,

$C=\{-3\}$

D) $x^2-3 x+2=0 \Rightarrow x=1,2$ and both are positive

So, this set has 2 elements, and thus it is not a singleton
Hence, the answer is the option (4).

Question 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) $\mathrm{S}=\{\phi\}$.

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) $\left\{\phi_{\}}\right.$is a singleton set with one element $\phi$.

Hence, the answer is the option (3).