Which of the following are examples of the singleton set?

(i) Integers $=\ldots-3,-2,-1,0,1,2,3, \ldots$

Given equation:

$x^{2}=4$

$\Rightarrow x=\sqrt{4}$

$\Rightarrow x=\pm 2$

If $x=-2$, then $x^{2}=(-2)^{2}=4$

If $x=2$, then $x^{2}=(2)^{2}=4$

So, there are two elements in a set.

$\therefore$ It is not a singleton set.

(ii) Integers $=-6,-5,-4,-3,-2,-1,0,1,2,3,4, \ldots$

Given equations:

$x+5=0$

$\Rightarrow x+5-5=0-5$

$\Rightarrow x=-5$

So, there is only 1 element in a given set.

$\therefore$ It is a singleton set.

(iii) Integers $=\ldots,-2,-1,0,1,2, \ldots$

Given equation: $|x|=1$

If $x=-1$, then $|x|=|-1|=1$

If $x=1$, then $|x|=|1|=1$

So, there are 2 elements in a given set

$\therefore$ It is not a singleton set.

(iv) Natural Numbers = 1, 2, 3, …

Given equation:

$x^{2}=16$

$\Rightarrow x=\sqrt{16}$

$\Rightarrow x=\pm 4$

$\Rightarrow x=-4,4$

but $x=-4$ not possible because $x \in N$

So, there is only 1 element in a set.

∴ It is a singleton set.

(v) Prime number = 2, 3, 5, 7, 11, …

Even Prime number = 2

∴ It is a singleton set.