Solution:
Option A: {x: x ∈ N and x‌2 < 36}
As we can see that, elements of {x: x ∈ N and x‌2 < 36} are: 1, 2, 3, 4, 5
So, there are 5 elements in the set {x: x ∈ N and x‌2 < 36}
Hence, the set {x: x ∈ N and x‌2 < 36} is a finite set.
Option B: {x ∈ z: 0 < x < 10}
As we can see that, elements of {x ∈ z: 0 < x < 10} are: 1, 2, 3, ........, 9
So, there are 9 elements in the set {x ∈ z: 0 < x < 10}
Hence, the set {x ∈ z: 0 < x < 10} is a finite set.
Option C: {x: x ∈ N and x‌2 = x}
∵ x ‌2= x
⇒ x‌2 - x = 0
⇒ x (x - 1) = 0
⇒ x = 0 or 1
But since x ∈ N. So, x = 0 ∉ {x: x ∈ N and x‌2 = x}
So, only x = 1 ∈ {x: x ∈ N and x‌2 = x}
Hence, {x: x ∈ N and x‌2 = x} is a finite set.
Option D: {x ∈ N: x is even}
As we can see that, elements of {x ∈ N: x is even} are: 2, 4, 6, 8, ........
So, there are infinitely many elements in the set {x ∈ N: x is even}
Hence, {x ∈ N: x is even} is not a finite set.
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