Determine Countability of a Set

Determine whether a given set is countable by demonstrating a one-to-one correspondence with the positive integers.

In this problem, you are asked to determine if a given set is countable by establishing a one-to-one correspondence with the positive integers. The concept of countability is a fundamental notion in set theory and real analysis, as it helps determine the size of infinite sets in a meaningful way. Countable sets are those that can be matched with the positive integers, providing a basis for comparing different types of infinities. This task requires a strong understanding of bijective functions, which are functions that pair each element of one set with one and only one element of another set, with no elements left unmatched in either set.

To approach this problem, it is crucial to first ensure that you understand the definitions of countable sets and bijections. When examining the given set, analyze and construct a rule or function that establishes a clear one-to-one mapping from the elements of the set to the natural numbers. Think about common techniques such as listing out elements systematically and identifying patterns or properties that facilitate the construction of a bijective function. This exercise not only deepens your comprehension of countable sets but also enhances your ability to work with functions and mappings in more complex analysis scenarios. The problem is an excellent illustration of how theoretical concepts are applied to make abstract definitions practically understandable.