9.1. Tutorial 5

Hello everyone welcome to tutorial number 5 so let us start with question number 1 we have to show in question number 1 that these two sets have the same cardinality. So the first set here is the set of all real numbers between (0,1) but excluding 0 as well as 1. Whereas the second collection here it is also the set of all real numbers between (0, 1] but 1 is inclusive that means 1 is allowed. That is why the square bracket here and 0 is not allowed.

In this chunk, we're introduced to the concept of cardinality through two different sets of real numbers. The first set contains all real numbers between 0 and 1, but does not include the endpoints 0 and 1. The second set contains all real numbers between 0 and 1, but it includes 1, while still excluding 0. The goal is to demonstrate that these two sets have the same cardinality. Cardinality is a measure of the 'size' of a set, which can be finite or infinite.

Imagine a fruit basket filled with fruits representing different real numbers. The first basket has all the fruits that are not the very first fruit (0) or the last fruit (1), while the second basket includes all except the first fruit but allows the last one (1). Even though the way we’ve defined what can be in these baskets is different, both baskets can actually contain the same number of fruits when we consider the variety of fruits we could list from one to the other.

So recall the Schroder-Bernstein theorem which says that, if you want to prove that two sets have same cardinality show injective mappings from the first set to the second and from the second set to the first.

The Schroder-Bernstein theorem is a fundamental principle in set theory that provides a method for demonstrating that two sets have the same cardinality. To apply this theorem, we need to establish two injective (one-to-one) functions: one mapping elements from the first set to the second set, and another mapping elements from the second set back to the first. If both mappings can be shown to be injective, it follows that both sets are of equal size or cardinality.

Think of a classroom with two groups of students, each with distinct roles. If you can create a buddy system where each student from the first group can be paired with a unique student in the second group, and vice versa, it shows that the number of students in both groups is the same. You’re demonstrating that both groups are balanced in size using these unique pairings.

Let us consider the injective mapping f which is the identity mapping so clearly this mapping is an injective mapping from this set to this set. Because you take any two different real numbers x and y the corresponding image will be x and y and they will be different and they will be in the range which is allowed as per function f.

In this chunk, we define the first injective mapping, referred to as the identity mapping. An identity mapping means that each element maps to itself. For any two distinct real numbers in our first set, the images in the second set remain distinct, thus satisfying the injective condition where different inputs yield different outputs. This mapping confirms that we can associate elements of the first set with distinct elements in the second set without overlap.

Imagine you have two different keys, each for a different door. The identity mapping would mean that Key A opens Door A and Key B opens Door B. Each key is unique, and if you have different keys, they will always correspond to different doors, just like how different real numbers map to themselves without any duplication.

Now if I want to take the injective mapping in the reverse direction then consider the injective mapping g defined to be x / 2 that means. If you want to find out the value of g(x) the output is (x / 2). So that means if your x here which we are considering is different from 1 then clearly that will fall in the range (0, 1).

This chunk discusses the second injective mapping, defined as g(x) = x/2. When we apply this mapping, we take each number from the second set and divide it by 2. If we consider numbers less than 1, they will remain within the range of the first set, maintaining the injective property, ensuring that different inputs continue to produce different outputs. Importantly, even the specific case of x = 1 produces a valid output (0.5), which still lies within the permitted range.

Consider a factory that produces different sizes of cakes. If you take a large cake (let’s say size x), and halve it (x / 2), the resulting smaller cake will still be less than 1, and every distinct cake size will create a new and distinct smaller size. This process maintains uniqueness, just like how different inputs produce different results in our mapping.

And it is again easy to verify that your mapping g is an injective mapping in the sense we have shown here two injective mappings so we can conclude that the cardinality of these two sets are the same.

At this point in the tutorial, we validate that both injective mappings have been successfully established. Since we have constructed two injective functions, one mapping from the first set to the second, and the other in reverse, according to the Schroder-Bernstein theorem, we can conclusively state that the two sets we started with have equal cardinalities. This illustrates a critical property in the field of mathematics regarding the size of infinite sets.

Returning to our classroom analogy, if we can pair every student in one group uniquely with students in another group, and we’ve shown both groups can do this without any ambiguities, we conclude both groups are equally sized. This equality holds true even if the classes are large enough to seem uncountable.

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