4.1.1. Lecture No # 28

Hello everyone welcome to this lecture on examples of countably infinite sets.

So just to recap in the last lecture we introduced the notion of countable and uncountable sets. Countable sets are those sets whose cardinality is either finite or whose cardinality is same as the set of positive integers. So the plan for this lecture is as follows. We will see several examples of countably infinite sets and we will also discuss some properties of countable sets specifically in the context of infinite sets.

In this chunk, the lecturer introduces the topic of countably infinite sets, which are sets that can be paired with the integers (like {1, 2, 3, ...}). The professor clarifies that countable sets can be either finite (like {1, 2, 3}) or infinite but structured like the set of positive integers. This sets the stage for the discussion on various examples and properties relevant to countable sets.

Think of a waiting line at a coffee shop. If there are 5 people (finite), you can easily count them. If the shop is always busy, with an endless line of customers coming in one after another (like the integers), that's similar to a countably infinite set. You can still assign each person a number like 1, 2, 3, ..., even if there is no end to the line.

So we first prove that the Cartesian product of the set of integers is a countable set. So again this might look non-intuitive, you have many elements in the Cartesian product of the set of integers compared to the set of integers itself because when I say that Cartesian product it is going to consist of all ordered pairs of the form (i, j) where i can be any integer, j can be any integer.

The lecturer points out that while the Cartesian product of integers, which consists of pairs (i, j), may seem larger than the set of integers itself, it can still be shown to be countable. This is because we can construct a method to list all the pairs without omitting any, thereby establishing a one-to-one correspondence with the positive integers.

Imagine a grid where each point represents a pair of integers, like a game board with coordinates. Just like you can systematically check each square on the board, you can also create a list of all possible pairs (i, j) by traversing this grid in a systematic manner.

So the idea is very clever here what we do is, So since we are considering the Set ℤ x ℤ it is nothing but the collection of all points in your 2 dimensional plane. Our goal is basically to give an enumeration of all the points in that infinite plane such that the enumeration should be well defined and we do not miss any point in the enumeration process.

In this segment, the focus is on how to enumerate points in the 2D integer plane represented by the set of Cartesian products given as ℤ x ℤ. The lecturer describes a systematic approach that starts at the origin (0, 0) and moves in a way that every integer pair gets listed, ensuring none are left out in the enumeration.

Consider a spider weaving a web. It methodically moves from the center to the outer edges, ensuring it touches every intersection on its path. Similarly, this enumeration strategy ensures every point in the integer plane is visited.

Now, we will see next whether the set of rational numbers which I denote by this ℚ notation is countable or not. Now intuitively it might look the answer is no because definitely rational numbers is a super set of the set of the integers.

Here, the lecturer raises a question regarding the countability of rational numbers (ℚ). Although it may seem logical that since rational numbers (which consist of fractions) include more numbers than integers, they might not be countable. However, the discussion pivots to proving that, through a clever enumeration technique based on previous concepts, it's possible to list all rational numbers without missing any.

Think of measuring ingredients for a recipe. Just as you can break measurements down into fractions like 1/2, 1/4, or 3/4, rational numbers allow for 'in-between' values. Even if there are countless ways to measure, we can still find a systematic approach to list every conceivable fraction.

Now let us consider the set of binary strings of finite length. What exactly that means? So, imagine a set Π consisting of 2 elements namely the element 0 and 1 and why 0 and 1? Because I am considering binary strings so, binary strings will be just a string of 0’s and 1’s.

This chunk introduces binary strings consisting exclusively of 0s and 1s. The lecturer explains how finite binary strings can be expressed and enumerated using a union of sets based on the length of these strings. This concept leads to the conclusion that the set of all finite binary strings is also countably infinite.

Consider texting using emojis. You can only use a selected set of emojis (like 0 and 1) in various combinations to form messages. Even if your texts can be of various lengths, starting from just 1 emoji to 10 or more, you can still list out every possible combination systematically.

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