Final answer:
The set Z × Z is shown to be countable by establishing a one-to-one correspondence between elements of Z × Z and the natural numbers using a defined pattern, often visualized as a spiral sequence around the origin where each integer pair is mapped to a unique natural number.
Step-by-step explanation:
To show that the set Z × Z (the Cartesian product of the set of all integers with itself) is countable, we can establish a one-to-one correspondence between the elements of Z × Z and the natural numbers (N), which are known to form a countable set. A common way to demonstrate this correspondence is by using Cantor's diagonal argument.
First, we list the ordered pairs in the following pattern: (0,0), (1,0), (0,1), (-1,0), (0,-1), (1,1), (-1,1), (1,-1), (-1,-1), (2,0), and so on, in a way that 'spirals' out from the origin. Notice how each ordered pair of integers can be assigned a unique position in this spiral sequence, creating a list where every pair will eventually appear.
To make it explicit, we can define a function f: Z × Z → N that assigns every pair (m,n) a unique natural number. For instance, if we describe the function as mapping (0,0) to 1, (1,0) to 2, (0,1) to 3, and so forth, every pair (m,n) corresponds to a natural number, thus proving that Z × Z is countable.