Explain why the following set is denumerable Your explanatio

Explain why the following set is denumerable. (Your explanation may be brief.)

{n³ | n N}

NOTE: Make sure to explain why the set is infinite, not just why it’s countable.

Solution

We know that an infinite set is denumerable if it is equivalent to the set of natural numbers.

Let A = {n³ | n N} and let f: N A be defined by f(n) = n³. Further, let a be an arbitrary element of A. Then n N such that a = n³ ( by definition of A). Then a = f(n). Hence f is surjective. Further, let a, b A such that f(a) = f(b). Then n₁ and n₂ N such that f(n₁) = a and f(n₂) = b ( as f is surjective). This implies that f(a) = n₁³ and f(b) = n₂³ ( by definition of f). Now, since f(a) = f(b), we have n₁³ = n₂³ . Further, since N is the set of natural numbers, hence n₁ and n₂ are both positive. Therefore, n₁ = n₂. This means that f is injective. Therefore f is bijective. Hence the set A = {n³ | n N} is denumerable.