9.1.4. Question 4

In question number 4 we want to show that if I take several countable sets and take their union namely if I perform union of countable number of countable sets then the resultant set is again countable.

This chunk introduces the main concept of the question, which is the union of countable sets. A countable set is one that can be listed in a sequence, such as the set of natural numbers or any finite set. The goal here is to prove that when you combine countable sets (even if you have an infinite number of them), the resulting set remains countable.

Imagine you have an infinite number of baskets, and each basket contains a countable number of apples. No matter how many baskets you have, if you combine all the apples into one big basket, you can still count them one by one. This illustrates that the total number of apples (the union of the sets) is still countable.

So since each of the sets S_i is countable it will have a listing of its own. So imagine that the listing of the elements of the set S_i is this. It is an infinite list, and the guarantee is that each element of the set S_i will eventually occur somewhere.

Each countable set, denoted as S_i, can be arranged in a sequential manner—like a list. For example, S_1 might contain elements like {1, 2, 3, ...}, S_2 might contain {a, b, c, ...}, and so on. The important point is that we can always create a list of all elements from these sets, ensuring that every element will appear at least once in the complete list.

Think of each S_i as different rows in a theater where each row contains chairs arranged for the audience. If the audience members (the elements in the set) fill these rows, we can always see that each chair gets filled, and everyone finds their seat, illustrating that every member will eventually be counted.

When I want to list out the elements of the set S, I will list down according to the pair of indices (i, j). And I will start by listing down all the elements of the form S_{i,j} such that the summation of the 2 indices is 2.

To create a complete list of all elements in the union of countable sets, a systematic approach is taken using pairs of indices. The elements are listed based on their index positions (i, j) where the sum of the indices is controlled to ensure all possible combinations are covered. This process ensures that we consider combinations starting from minimum index sums and gradually include more pairs. This systematic listing guarantees that no element is missed.

Imagine a library where books are organized by their genre (i) and author (j). When you want to list all books, you first look for all books by a specific genre and author whose names begin with the same letter, ensuring you get a complete collection without skipping any.

Because you take any element x belonging to the bigger set S, x will belong to some S_i.

This point emphasizes that every element in the union of the countable sets must belong to at least one of the individual countable sets. Thus, as we construct our comprehensive list, we ensure that every possible element is included by confirming that they originate from the initial countable sets. As a result, the union of these sets remains countable.

Think of a classroom where every student must belong to at least one club. If you collect a list of all students from various clubs combined, you ensure that every student appears on your list—the more clubs you add, the more inclusive your total list becomes, without omitting anyone.

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