Lesson 1

The set is called countable if it is in one-to-one correspondence with \(\mathbb{Z}_{> 0}\). In other words, the set is countable when its elements can be enumerated: \(x_{1}, x_{2}, \dotsc\).

The set \(\mathbb{Z}\) is countable: its elements can be enumerated as follows: \[0, 1, -1, 2, -2, 3, -3, 4, -4, \dotsc\] To show that a set is countable, it is enough to list its elements in an infinite sequence. For example, the set \(\mathbb{P}\) of prime numbers is countable, because we can take the set of all positive integers and cross out all non-prime numbers—this will result in an infinite sequence of primes.