(5) Show that the set \( E \) of even positive integers is countable.

To show that the set \( E \) of even positive integers is countable, we can construct a function that establishes a one-to-one correspondence between \( E \) and the set of positive integers \( \mathbb{N} \). Consider the function \( f: \mathbb{N} \rightarrow E \) defined by \( f(n) = 2n \). This function takes each positive integer \( n \) and maps it to an even integer. Now, let's see if this function is bijective. For every \( n \in \mathbb{N} \), \( f(n) \) produces a unique even integer because multiplying by 2 yields distinct outputs for different inputs. Also, for each even positive integer \( e \), there exists a positive integer \( n \) such that \( e = 2n \). This shows that every even positive integer is accounted for, confirming that \( f \) is onto. Since we've demonstrated that there is a bijection between \( \mathbb{N} \) and \( E \), we conclude that the set of even positive integers \( E \) is countable. Thus, we have successfully shown that the even positive integers can be paired with the natural numbers.