State which of the following sets are singleton sets.
$(i) A = {x | x = 16}$
$(i i) B = {y | y^{2} = 36}$
$(i i i) C = {p | p \in I, p^{3} = - 8}$
$(i v) D = {q | (q - 4)^{2} = 0}$
$(v) E = {x | 1 + 2 x = 3 x, x \in W}$

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Solution

Singleton set: A set that contains a single element is called a singleton set.
$(i) A = {x | x = 16}$
Roster form of set $A = {16}$
This set consists of only one element. Therefore, it is a singleton set.
$(i i) B = {y | y^{2} = 36}$
Since, $y^{2} = 36$
$\Rightarrow y = \pm 6$
Roster form of set $B = {+ 6, - 6}$
This set consists of two elements. Therefore, it is not a singleton set.
$(i i i) C = {p | p \in I, p^{3} = - 8}$
Since, $p^{3} = - 8$
$\Rightarrow p^{3} = (- 2)^{3}$
$\Rightarrow p = - 2$
Roster form of set $C = {- 2}$
This set consists of only one element. Therefore, it is a singleton set.
$(i v) D = {q | (q - 4)^{2} = 0}$
Since, $(q - 4)^{2} = 0$
$\Rightarrow q - 4 = 0$
$\Rightarrow q = 4$
Roster form of set $D = {4}$
This set consists of only one element. Therefore, it is a singleton set.
$(v) E = {x | 1 + 2 x = 3 x, x \in W}$
Since, $1 + 2 x = 3 x, x \in W$
$\Rightarrow 2 x - 3 x = - 1$
$\Rightarrow - x = - 1$
$\Rightarrow x = 1$
Roster form of set $E = {1}$
This set consists of only one element. Therefore, it is a singleton set.

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