Yes, the set Z of all integers is countable.
To understand why, let's define what it means for a set to be countable. A set is countable if its elements can be put into a one-to-one correspondence (bijection) with the set of natural numbers (N = {1, 2, 3,...}). This means we can create a numbered list of all the integers without missing any and without any repetitions (although the numbers on the list need not be in increasing order).
We can explicitly demonstrate this countability by arranging the integers in a sequence like this:
0, 1, -1, 2, -2, 3, -3, 4, -4, ...
This arrangement maps each natural number to a unique integer. We can define a function f from the set of natural numbers N to the set of integers Z as follows:
- If n is even, then f(n) = n/2.
- If n is odd, then f(n) = -(n-1)/2.
This function is a bijection, demonstrating that we can map each natural number to exactly one integer and vice versa. Therefore, since we can create a bijection between the natural numbers and the integers, the set of integers is countable. Specifically, because the set of integers is not finite, it is countably infinite.
