Question

Prove that the set \(A=\\{\ln (n): n \in \mathbb{N}\\} \subseteq \mathbb{R}\) is countably infinite.

Short answer

The set \(A=\{\ln (n): n \in \mathbb{N}\}\) is countably infinite because there is a one-to-one correspondence between the set \(A\) and the set of natural numbers, \(\mathbb{N}\). This correspondence is given by the function \(f(n) = \ln(n)\), which maps each natural number \(n\) to \(\ln(n)\) in \(A\).

Step-by-step solution

Understanding the definitions

The set of natural numbers \(\mathbb{N}\) is defined as a countable set. A set is countable if there exists a bijective function (a one-to-one correspondence) from the set of natural numbers to the set. The real numbers \(\mathbb{R}\) are uncountable, but the set \(A=\{\ln (n): n \in \mathbb{N}\}\) is a subset of real numbers and this fact does not necessarily make it uncountable.

Finding a bijective function

In order to prove that the set A is countable, we need to find a bijective function between the set A and the set of natural numbers \(\mathbb{N}\). Let's define a function \(f: \mathbb{N} \rightarrow A\) that maps each natural number \(n\) to \(\ln(n)\). This function is clearly bijective: for every natural number \(n\), there is one and only one \(\ln(n)\), and every \(\ln(n)\) comes from one and only one \(n\). Hence, every element in A can be paired with a unique element in \(\mathbb{N}\), and vice versa.

Concluding that A is countably infinite

Therefore, the function \(f\), which establishes a one-to-one correspondence between the set \(\mathbb{N}\) of natural numbers and the set \(A\), proves that the set \(A\) is countable. As \(A\) has an infinite number of elements (since there are infinitely many natural numbers), we conclude that \(A\) is countably infinite.