Question

Define a singleton set with an example is $A=${Set of even prime numbers} a singleton set?

Answer 1

Hint: Here, we have to define the singleton set.
And also, I have to find whether the given set is a singleton set or not.
For that first we have to simplify the given set as the given set is written in words and we have to find the number for it.
If the simplified given set contains only one number then. It will be a singleton set but if the simplified given set contains more than one number then it will not be a singleton set.

Complete step-by-step solution:
Definition of singleton set:
There are so many types of sets.
Singleton set is one of them and it is defined as,
When a set shares only one term or element or number, then it is known as a singleton set.
The example for a singleton set is as.
$A=$ $\{y:y$ is a whole number which is not a natural number}
i.e. $A=\left\{0\right\}$
It means that, the set $A,$ has only one number because the number which natural as well as whole is only one and that is $0$ (zero)
Now, the given expression is that,
$A=${Set of even prime number}
We have to find whether it is singleton or not.
We know that, the even numbers as it is follows,
$2,4,6,8,...$ and so on.
The prime number means. It can be factorized only from two factors : one and itself.
In even number set there is only number $2$ which has only two factors as $1$ and itself $2.$
Another number has factors more than two. For example, $4$ can be factorize as $1,2$ and $4;6$ can be factorize as $1,2,3$and $6.$
$\because 2=1,2$
Therefore, $2$ is the only prime even number.
$A=${Set of even prime number}
$A=\left\{2\right\}$
As the given set $A$ contains only one number therefore it is a singleton set according to its definition.

Note: The set is nothing but the place where we collect the items or objects, within the curly brackets.
These items or objects are also called as elements of a set. Generally, the set is the collection of apparent (distinct) objects of the same type.
As per number of element the type of set are classified as below:
(1) Empty set: The set which does not contains any number or element known as empty set for e.g. $P=\{x:s$ is a leap year between $1904$ and $1908\}$
As there is no leap year between $194$ and $1908$
$\therefore P=\left\{\right\}$ or $P=\varphi$
(2) Singleton set: The set which contains only one number is known as a singleton set.
For e.g. $A=\{y:y$ is whole number which is not a natural number}
(3) Finite set: The set which has no number or element or a definite number then is called a finite set.
For e.g. $A=\{x:x$ is a month in a year}
As there are $12$ element in set $A$
The empty set can also be called a finite set.
(4) Infinite set: The set which contains an infinite number or elements is known as an infinite set.
For e.g. $A=\{x:x$ is a natural number}
As there are infinite natural numbers.