9.1.6. Question 6

In part a, you want to find out whether the set of integers divisible by 5 but not by 7 is countable or not. So first of all what exactly is this set so let me call, denote this set by S. So the set S will have the number 0 it will have +5 – 5 it will have +10 – 10 and so on. It would not have 35 it would not have + 35 it would not have -35 because 35 is divisible by 5. But it is also divisible by 7 so are not supposed to include multiples of 7. We are not supposed to include multiples of both 5 and 7; so now we want to show whether the set S is countable or not. So definitely S is an infinite set but the question is can we enumerate the elements of the set S. And it is easy see that we can always list down the elements of the set according to their absolute values. And if I focus on the absolute values then both +x and –x will take the same absolute value x. So what I will do is in the listing I will write down +x and –x so basically what I am saying is that my listing here is 0 followed by +5 followed by -5 then +10 followed by -10 and so on. You cannot do the following: you cannot say that first list down all the positive things up to infinity and then followed by the minus things. This is not a valid listing because if we do this then we do not know whether we will come back ever and start listing down the negative numbers here. Because when we start going towards the positive multiples of 5 in infinity we will never stop and there is no coming back. So that is why it is only the first listing which is a valid listing not the second one.

The set S contains integers that are divisible by 5 but not by 7. This means that the set includes values like 0, 5, 10, -5, -10, etc., but excludes multiples of 35 (5 and 7's product). Since this set includes both positive and negative values and continues indefinitely in both directions, it is infinite. We can enumerate the members of S by listing them as follows: 0, 5, -5, 10, -10, etc. Importantly, while we can create a sequence, we cannot just list all the positive integers first and then the negative ones because that would make us lose the order needed to cover all elements. Thus, S is countable, meaning we can list its integers in a structured manner.

Think of it like organizing a collection of books. You have books that belong to specific genres. Imagine if you decide to list them by genre, you would list all the books of one genre before moving onto another. Now, if you instead said, 'I will list all the fiction books first and then the non-fiction books next,' you might end up forgetting some books or leaving them out because you get caught up in one genre. The same principle applies here when it comes to ensuring we enumerate S correctly.

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. And it turns out that this set is countable because I can view it as union of several countable sets okay. So let S denotes 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 that means you have a series of infinite ones and occurring again and again. Followed, by the real number 0.1 followed by real number 0.11 followed by real number 0.111 and so on. 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.

This part focuses on determining whether the subset of real numbers that only have the digit 1 in their decimal representation is countable. The creator of the set, S, breaks it down into smaller sets, where each smaller set corresponds to numbers with a different number of 1s before the decimal. Each of these smaller sets is countable, and by taking their union, we still have a countable set since the union of countable sets remains countable. This shows that even though our focus is limited to numbers with 1s, they can still be listed systematically.

Imagine a bakery that only bakes pastries with a single type of topping, say chocolate. If the bakery decides to make different styles of pastries with that one topping, it can create a systematic layout. Think of these styles as different countable sets. If there’s a list for each style, then the overall collection of chocolate pastries is organized and countable. It's just like listing numbers based on the number of 1s in their decimal representation.

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