4.3. General Results About Cardinality

The first theorem is that if you have 2 sets A and B and if they are countable then their union is also countable. So I am not saying anything about the number of elements in the union A and B. Of course, what I am saying is that it is always possible to list down the elements of A U B. So how we are going to prove it?

This theorem states that if we have two sets, A and B, which can be counted (i.e., they are countable), then the combination of these sets (their union) can also be counted. To understand this, think of countable sets like lists of items you can enumerate, such as a list of favorite books or movies. When you take two such lists (A and B), it's possible to create a new list that contains every item from both original lists (A U B). What's important is not necessarily how many items are in the new list but that we can find a way to list them all sequentially. The proof can vary depending on if A and B are finite or infinite, but it fundamentally hinges on the fact that countable sets can be organized sequentially.

Imagine organizing two boxes of toys where Box A contains 5 toys and Box B contains 3 toys. If you write down the names of all the toys from Box A and then follow with the names of all the toys from Box B, you've created a new list that includes all toys from both boxes. You can count them all in one go, demonstrating the principle of combining countable sets.

Now we can have various cases depending upon whether A and B are countably finite or countably infinite. Case 1 when both A and B are finite that means say if the cardinality of A is m and the cardinality of B is n then it is easy to see that the cardinality of union of A and B will be m + n which is a finite number and hence A U B is also countable.

There's a distinction in our proof based on whether the sets are finite or infinite. If both sets, A and B, have a finite number of elements (let's say A has m elements and B has n elements), then counting the total number of distinct elements in A combined with all the distinct elements in B just gives us m + n, which is also a finite number. Therefore, we can conclude that their union A U B is countable, as it retains a finite cardinality.

Think of having 5 apples in one basket (A) and 3 oranges in another basket (B). If you put them all in a single basket (A U B), you have a total of 5 + 3 = 8 pieces of fruit. You can count them all easily, affirming that the union of your fruits (A U B) is still a manageable, countable collection.

Case 2 is when exactly 1 of the set A and B is finite whereas the other set is countably infinite. Now again we can have 2 possible cases depending upon which of the 2 sets is countably infinite. So what we can do is we can list down the elements of A U B as follows. First list down the elements of B which are finite in number, m in number followed by the elements of the set A.

When one set is finite and the other set is countably infinite, we can still create a sequence for their union. Let's say A is infinite and B is finite. To list the elements of A U B, we start with the finite set B (because we can specifically point out where each finite element is going to be in the count), followed by the infinite elements of A. This approach is valid because it ensures that the finite elements are well-defined and we won’t lose track of where they exist in the larger sequence.

Imagine a scenario where you have an infinite playlist of songs (set A) and a few specific new songs (set B). When making a new playlist (A U B), you could first add the 5 new songs (a definitive, small number) and then continue with the infinite playlist of older songs. The new songs are easily locatable at the beginning, and you know that as you continue, the playlist plays on indefinitely.

The third case is when both A and B set are infinite and countable, because I am assuming my A and B sets are countable and if A; and B are both infinite that means both the cardinality of A as well as the cardinality of B is א . And I want to show that A U B is also countable by giving you a valid sequencing for the elements in the union of A and B.

In the scenario where both sets A and B are infinite, they each possess a countable infinity of elements. We demonstrate their union is still countable by producing a valid enumeration for A U B. A common method is to alternate between elements from A and B—take the first element from A, then from B, repeat for the second, and so forth. This ensures that every element from both infinite sets is represented completely, allowing us a clear and structured way to reference every item in their union.

Imagine two never-ending lines of people (one for A and one for B). To ensure everyone gets counted, you alternate between taking a person from the first line (A) and then one from the second line (B). Even though both lines are infinite, your counting process maintains clarity and organization because you can always identify whose place is where in the combined queue.

Now second interesting result about the cardinality theory is what we called as the Schroder Bernstein theorem which says the following. If you have 2 sets such that the cardinality of A is less than equal to cardinality of B and simultaneously the cardinality of B is less than equal to the cardinality of A. Then we can conclude that both set A and B have the same cardinality.

The Schroder-Bernstein theorem provides a powerful way of determining when two sets have the same number of elements, also known as the cardinality. If we can find a way to injectively map (essentially translate) elements from one set to another (say A to B), and also from B back to A, then we can confidently assert that both sets must be the same size. This principle highlights the interconnectedness between different sets and how their relationships impact how we understand counting.

Imagine two classrooms (A and B) where the number of students in each is unclear, but we can observe and pair students from each room towards teaching each other. If every student in room A can teach a student in room B (mapping A to B), and with a similar mapping back, we would conclude that both classrooms have an equivalent number of students, despite not actually counting them physically.

Now the third result about the cardinality is the following. If I take any subset of a countable set then it should be also countable. So, there are 2 cases the above statement is obviously true if the set A is a countably finite set.

This principle states that any time we derive a subset from a countable set, that subset must also be countable. For instance, if we think of a set as a bag of marbles that can be counted, any particular selection of marbles (a subset) will still remain countable no matter how many you take. This provides a consistent foundation for understanding how size and count interact, reinforcing the notion that smaller parts derived from larger sections will maintain the property of being countable.

Consider a bag filled with 20 different kinds of candies (set A). If you decide to put only 5 types into a smaller bag (subset B), even though you've taken some out, you can still count the candies in the smaller bag. It ensures that even as you reduce the overall count, it remains manageable and countable.

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