3.6. Examples of Countably Finite Sets

So now based on these definitions let us see some examples of countably finite sets. So I start with the set of odd positive integers and let me denote the set of odd positive integers by this notation ℤ+. And my claim is that the set of what positive integers is the countable set. So the statement might look very non-intuitive because definitely you have more integers than the set of odd positive integers right.

The set of odd positive integers, denoted as ℤ+, is a countable set. While it may seem non-intuitive to think that there are fewer odd integers than all integers, it is important to understand that countability focuses on the ability to establish a one-to-one correspondence (bijection) between the set of odd positive integers and the set of positive integers. This means that each odd positive integer can be matched with a unique positive integer.

Think of the set of odd positive integers as a unique queue at a bakery, where every odd-numbered person represents an odd integer. If you compare it to a larger queue containing everyone, including both even-numbered and odd-numbered people, it may feel like the odd queue is smaller. However, for every odd person in this queue, there is an exact match (the half of the queue) in the larger group, proving they are equal in number despite their appearances.

So the bijection is as follows. So consider the sequence 1, 3, 5, 7 like that where the nth term is the sequence is 2n – 1 and so on. So clearly this sequence is sequence of infinite positive integers and each number in this sequence is odd. My claim is that this is a valid sequence for the set of odd positive integers it is valid in the sense no element in this sequence is repeated.

To establish that the set of odd positive integers is countable, we can define a mapping (bijection) from the set of positive integers to the set of odd positive integers using the function f(n) = 2n - 1. This function generates all odd numbers starting from 1, 3, 5, etc., ensuring that for every positive integer n, there corresponds an odd positive integer, and vice versa. The sequence generated does not repeat elements, fulfilling the criteria for a valid sequence.

Picture this as creating a unique seating arrangement for odd-numbered guests at an event. Every guest with an odd number (1st, 3rd, 5th, ...) is mapped to a seat at the table (where the seats start at 1). As each guest takes their respective seat, we see that every odd-numbered guest sits down without any duplications, illustrating how we can perfectly pair each guest with a seat in a one-to-one mapping.

Now what we are going to show next is a very interesting fact we are going to show here that the set of integers is a countable set. Mind it the set of integers has both positive integers as well as negative integers. But what we are now going to show is that cardinality wise the set of integers is same as the set of odd positive integer, they have the same number of elements.

The set of all integers, both positive and negative, can also be counted in the same way as the set of odd positive integers. To represent this set, we can list the integers in an alternating sequence: 0, 1, -1, 2, -2, 3, -3, etc. This allows all integers to be accounted for without missing any, thereby showing that even though it may appear larger, the set of integers is countable as it can be paired bijectively with the set of positive integers.

Imagine sitting in a two-sided waiting room where people with positive room numbers (1, 2, 3, ...) sit on one side, while those with negative room numbers (-1, -2, -3, ...) sit on the other side. By counting the seats from 0 alternately (first seating a positive number then a negative), every patient can be accounted for, illustrating the countable nature of integers, even if at first glance it appears more complex.

Next let us consider the set of prime numbers and for those who do not know what is a prime number? A number p is prime provided it is divisible by only 1 and p; that means there are no other divisors for the number p apart from the number 1 and the number p.

The set of prime numbers, which consists of numbers like 2, 3, 5, 7, etc., is also countable. Just like other sets we’ve discussed, we can establish a bijection to the set of positive integers by simply listing the prime numbers in increasing order. This list continues infinitely, demonstrating that the primes do not have a finite count, but they can be arranged in a sequence indexed by positive integers, thus making them countable.

Think of prime numbers as treasures hidden in a vast treasure chest, where the first few treasures reveal themselves distinctly (like the first few primes: 2, 3, 5, 7, ...). Although there are infinitely many treasures, if you keep listing them, each can be located by following the increasing order of their discovery, showing that they can be counted one by one, just like the odd integers or full integers.

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