Which of the following is a singleton set?
A) $\left\{ {x:\left| x \right| = 5,x \in N} \right\}$
B) $\left\{ {x:\left| x \right| = 6,x \in Z} \right\}$
C) $\left\{ {x:{x^2} + 2x + 1 = 0,x \in N} \right\}$
D) $\left\{ {x:{x^2} = 7,x \in N} \right\}$

Hint: In the given question, we are given a few sets in the options and we need to find which of them is a singleton set. Set is defined as a well-defined collection of objects. These objects are referred to as elements of the set. If a set contains only one element, then it is called a singleton set. For example: $A = \left\{ {x:x{\text{ }}is{\text{ }}an{\text{ }}even{\text{ }}prime{\text{ }}number} \right\}$. Now, we will check for each option individually whether the given set is a singleton set or not.

Complete step by step answer:
Let’s analyse the given options.
A) $\left\{ {x:\left| x \right| = 5,x \in N} \right\}$
$ \to \left| x \right| = 5$
Therefore, we get
$ \Rightarrow x = \pm 5$
$\therefore x \in N$
As we know, natural numbers refer to a set of all whole numbers excluding $0$. Natural numbers are also known as non-negative integers.
So $x$ will assume only one value which is $5$. Therefore, this is a singleton set.

B) $\left\{ {x:\left| x \right| = 6,x \in Z} \right\}$
$ \to \left| x \right| = 6$
Therefore, we get
$ \Rightarrow x = \pm 6$
$\because x \in Z$
As we know, an integer is a number with no decimal or fraction part. It is a special set of whole numbers composed of zero, positive and negative numbers. $x$ can assume two values $6$ and $ - 6$.
Therefore, this is not a singleton set.

C) $\left\{ {x:{x^2} + 2x + 1 = 0,x \in N} \right\}$
$ \to {x^2} + 2x + 1 = 0$
On completing the square, we get
$ \Rightarrow {\left( {x + 1} \right)^2} = 0$
Therefore, we get
$ \Rightarrow x = - 1$
$\because x \in N$
As we know, natural numbers refer to a set of all whole numbers excluding $0$. Natural numbers are also known as non-negative integers.
So this is a null set.

D) $\left\{ {x:{x^2} = 7,x \in N} \right\}$
$ \to {x^2} = 7$
Therefore, we get
$ \Rightarrow x = \pm \sqrt 7 $
Which is an irrational number. Natural numbers refer to a set of all whole numbers excluding $0$. Natural numbers are also known as non-negative integers.
But $x \in N$. So this is a null set.

Therefore, option (A) is correct.

Note:
To solve this type of question, one must know properties of different types of sets. The different types of sets are as follows:
Empty set: A set which doesn’t have any elements is called an empty set. For example: A set of bananas in the basket of apples is an example of an empty set because there are no bananas present in the basket.
Finite set: A set which consists of a definite number of elements is called a finite set. For example: A set of whole numbers up to $10$. $A = \left\{ {0,1,2,3,4,5,6,7,8,9,10} \right\}$.
Infinite set: A set which is not finite is called an infinite set. For example: A set of all natural numbers. $A = \left\{ {1,2,3,............} \right\}$.