9.5. Set of Real Numbers with Decimal Representation of Only 1s

In part B we want to find out whether the set of all real numbers whose decimal representation consist of only 1s is countable or not. So remember the set of real numbers is uncountable we proved that but here we are not focusing on the set of all real numbers. We are focusing only on the real numbers whose decimal representation have only 1s and no other digit.

This chunk introduces the topic of real numbers whose decimal representations consist solely of the digit 1. It brings attention to the fact that while the set of all real numbers is uncountable, the focus here is on a more specific subset that has unique properties.

Think of this like focusing on a particular group of people in a very large city (all of whom are in some way unique), where you are only interested in those who wear a specific type of clothing (in this case, the clothing being the digit '1' in decimal format).

And it turns out that this set is countable because I can view it as union of several countable sets okay. So let S denote the real numbers consisting of only 1s where you do not have anything before the decimal point. And after the decimal point you list down all the numbers starting with real number where you have recurring 1s...

This chunk explains that the specific set S can be considered a union of several smaller sets, each of which is countable. By defining S in terms of its decimal representations—specifically those that consist only of the digit '1'—the concept of countability can be effectively illustrated. By considering different ways to construct numbers starting with 1s, it implies there are many distinct yet countable variations.

Imagine you’re collecting tickets to different movie screenings only for films that start with the letter 'A.' Each screening can be thought of as a different way to write a number in set S (like '1.1', '1.111', etc.), which are all countable since you can create a list of them.

Remember 0.1 is not same as the real number where 1 is recurring. 0.1 you do not have anything after the first 1 here. But 0.1 with the recurring one denotes this real number which is clearly different from 0.1. So these are the elements in my set S and I have a valid listing here.

In this part, there is a careful distinction made between two types of numbers: '0.1' and '0.111...' (with 1s recurring). This differentiation is crucial because it shows that even within the subset of numbers with only 1s, there can be different representations that illustrate the number of ways we can formulate elements of the set. Creating a valid listing assures us that all members can be enumerated, confirming that the set is countable.

Consider buying books at a bookstore. Just like '0.1' is a standalone title that may be different from a series of '1's, one book may be a single title while another represents many volumes of a series. Both can be listed individually at the store, helping us understand that we can effectively count how many titles— or representations—are present.

That means now I now have a valid listing here and bigger set S is the union of several countable sets and hence the bigger set S will be countably infinite.

This final chunk concludes that since set S can be expressed as the union of these smaller, countable sets, the entire collection remains countable (and thus countably infinite). This conclusion reinforces a fundamental property in set theory: the union of a countable number of countable sets is countable.

Think about a library system where each small bookshelf represents a different countable set of books. Each shelf has a finite number of books (countable), and together they fill a library. The entire library can still be counted even though it contains an infinite number of shelves. This mirrors how the union of countable sets yields a complete countable whole.

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