Which of the following are examples of the singleton set?
(i) {x : x ϵ Z, x2 = 4}.
(ii) {x : x ϵ Z, x + 5 = 0}.
(iii) {x : x ϵ Z, |x| = 1}.
(iv) {x : x ϵ N, x2 = 16}.
(v) {x : x is an even prime number}
(i) Integers = …-3, -2, -1, 0, 1, 2, 3, …
Given equation:
x2 = 4
⇒ x = √4
⇒ x = ± 2
If x = -2, then x2 = (-2)2 = 4
If x = 2, then x2 = (2)2 = 4
So, there are two elements in a set.
∴ It is not a singleton set.
(ii) Integers = -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, …
Given equations:
x + 5 = 0
⇒ x + 5 – 5 = 0 – 5
⇒ x = -5
So, there is only 1 element in a given set.
∴ It is a singleton set.
(iii) Integers = …, -2, -1, 0, 1, 2, …
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
∴ It is not a singleton set.
(iv) Natural Numbers = 1, 2, 3, …
Given equation:
x2 = 16
⇒ x = √16
⇒ x = ± 4
⇒ x = -4, 4
but x = -4 not possible because x ∈ 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.