This question from set theory asks about the cardinality (a measure of the number of elements) of the set of rational numbers, \( \mathbb{Q} \).
Step 1: Understanding the Question:
We need to determine if the set of all rational numbers can be put into a one-to-one correspondence with the set of natural numbers (\(\mathbb{N} = \{1, 2, 3, \dots\}\)). If it can, the set is countable; otherwise, it is uncountable.
Step 2: Key Formula or Approach:
The approach is to demonstrate, or recall the demonstration, that a systematic listing of all rational numbers is possible. This is famously achieved through Cantor's diagonalization argument.
Step 3: Detailed Explanation:
A set is countable (or countably infinite) if its elements can be listed in a sequence, like \(r_1, r_2, r_3, \dots\).
The set of rational numbers, \( \mathbb{Q} \), consists of all numbers that can be expressed as a fraction \(p/q\), where \(p\) is an integer and \(q\) is a non-zero integer.
Although between any two rational numbers there is another rational number (the set is dense), the entire set is surprisingly countable.
We can construct a list of all positive rational numbers by arranging them in a grid where the entry in row \(q\) and column \(p\) is the fraction \(p/q\). We can then traverse this grid diagonally, adding each new fraction to our list. This process ensures that every possible rational number will eventually be included in the sequence. By extending this to include 0 and negative rationals, we can list all elements of \( \mathbb{Q} \).
Since such a list can be constructed, the set of rational numbers is countable.
Step 4: Final Answer:
The set of all rational numbers \( \mathbb{Q} \) is a countable set.