3.7. Set of Odd Positive Integers

Let me 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.

This chunk introduces the concept of odd positive integers, represented by the symbol ℤ+. The speaker claims that the set of odd positive integers is a countable set. This may seem surprising because it feels like there are more integers than just the odd ones. However, the idea of countability implies that we can establish a relationship between the odd positive integers and another known set, even if it seems at first like one set contains more elements than the other.

Imagine a large basket filled with apples. If I tell you that I can take out just the red apples and I still have a way to number them all, you might initially think that the original basket of apples (red and green) is much larger, but as we count and find a systematic way to list the red apples, we discover that while they are fewer in variety, they can still be counted in a similar manner.

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.

In this chunk, the speaker describes how to create a sequence of odd positive integers by the formula 2n - 1, where n is a positive integer. This produces the sequence: 1, 3, 5, 7, ..., which consists solely of odd positive integers. By establishing this sequence, the speaker demonstrates that for every positive integer n, there corresponds exactly one odd positive integer, which helps to show the concept of countability through bijection.

Think of a numbering system for seating in a theater where only every first or odd seat is considered valid. Each seat number corresponds to an odd positive integer. For instance, seat 1, 3, 5, etc., mimic how odd integers relate to natural number positions. Thus, even if you know there are more total seats, you can still effectively manage and audit only the odd-numbered ones.

It is very easy to see that the function is injective because if you take the integers 2n-1 and 2m-1 and if they are same then that is possible only if your n is equal to m and clearly the function is surjective as well.

This part of the section explains that the mapping from the set of positive integers to the set of odd positive integers is both injective and surjective. This means that each n results in a unique odd integer (injective), and every odd integer corresponds to some n (surjective). This one-to-one correspondence illustrates that both sets have the same cardinality, hence they are both countable.

Consider a school with two kinds of lockers. The odd-numbered lockers can represent the odd integers, and the total number of students represents the positive integers. Each student gets exactly one odd-numbered locker, ensuring each odd locker is used, illustrating the injective aspect. If every odd-numbered locker has a corresponding student, we also see the surjective aspect clearly.

So this shows that cardinality wise the set of odd positive integers is the same as the set of positive integers. Even though, intuitively you have more elements in the set of positive integers than the set of odd positive integers.

Here, the speaker concludes that the cardinality of the set of odd positive integers is indeed the same as that of the set of positive integers, despite the initial assumption that there are more positive integers overall. The conclusion is drawn from the established bijection, which shows that there is an equal number of odd positive integers and positive integers, fundamentally changing our understanding of 'more' in the case of infinite sets.

Think of it like having an infinite forest with trees where you have labeled just the tall trees and just the short trees. Even if intuitively we might think there are more tall trees, by counting them in a systematic manner (like numbering them), we can establish that there are just as many tall trees as all the trees in the forest, demonstrating the profound nature of infinity.

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