Show that the set Z+ × Z+ is countable

To show that the set Z+ × Z+ is countable, we can provide a bijection (one-to-one correspondence) between the set of natural numbers N and the set Z+ × Z+.

Here are the steps:

1. Consider the set Z+ × Z+ as a two-dimensional grid of ordered pairs of positive integers.

2. We can traverse this grid in a diagonal manner. Start from the point (1,1), then move to (2,1), then (1,2), then (3,1), then (2,2), then (1,3), and so on.

3. In this way, we can assign a unique natural number to each ordered pair in the set Z+ × Z+. For example, (1,1) is assigned 1, (2,1) is assigned 2, (1,2) is assigned 3, and so on.

4. This process can be continued indefinitely, and every ordered pair in the set Z+ × Z+ will eventually be assigned a unique natural number.

5. Therefore, we have established a bijection between the set of natural numbers N and the set Z+ × Z+.

6. Since the set of natural numbers N is countable, this implies that the set Z+ × Z+ is also countable.

This completes the proof.