Show that the set of positive rational numbers is countable.

Sure, here is a step-by-step proof that the set of positive rational numbers is countable:

1. First, we need to understand what a countable set is. A set is countable if its elements can be put into one-to-one correspondence with the set of natural numbers. In other words, if we can list out all the elements of the set in a sequence, then the set is countable.

2. The set of positive rational numbers is the set of all positive fractions, i.e., numbers of the form p/q where p and q are natural numbers.

3. We can list out these numbers in a two-dimensional grid where the row number represents the numerator and the column number represents the denominator. The grid would look something like this:

   1/1 1/2 1/3 1/4 ... 2/1 2/2 2/3 2/4 ... 3/1 3/2 3/3 3/4 ... 4/1 4/2 4/3 4/4 ... ...

4. Now, we can traverse this grid in a diagonal manner such that every fraction is included in the sequence. The sequence would look like this:

   1/1, 2/1, 1/2, 1/3, 2/2, 3/1, 4/1, 3/2, 2/3, 1/4, ...

5. This sequence includes every positive rational number exactly once, so we have established a one-to-one correspondence between the set of positive rational numbers and the set of natural numbers.

6. Therefore, the set of positive rational numbers is countable.