9.4.1. Set S Definition

We have to show that these two sets have the same cardinality. The first set is the set of all real numbers between (0, 1) but excluding 0 and 1. The second set is the set of all real numbers between (0, 1] where 1 is included.

In this section, we are trying to establish that the cardinality (size) of two sets is the same. The first set contains numbers strictly between 0 and 1, which means neither endpoint is included. The second set, however, includes 1, making it slightly larger, but we will prove that they actually have the same size through a mathematical principle called the Schroder-Bernstein theorem.

Think of two swimming pools where Pool A has all the people in the pool except for the first and last person, while Pool B has all the people in the pool including the last one. We suspect that both pools might hold the same number of people, even if one seems slightly fuller. To find out for sure, we can use a method to show there's a way to match the swimmers from Pool A to Pool B and vice versa.

To prove that these two sets have the same cardinality, we refer to the Schroder-Bernstein theorem. This theorem states that if there are injective mappings (one-to-one functions) from set A to set B and from set B to set A, then the two sets have the same cardinality.

The Schroder-Bernstein theorem tells us that if we can find a way to map each element of one set to another uniquely, while also mapping the second set back to the first uniquely, then both sets are the same size in terms of cardinality. This makes it essential for our proof because we need to show these mappings exist.

Imagine pairing students between two classrooms. If students in Classroom A can be paired uniquely with students in Classroom B, and then if we can somehow pair back the students in Classroom B uniquely with those in Classroom A, we can conclude both classrooms have the same number of students, even if one appears bigger.

Let us consider the injective mapping f which is defined as the identity function, meaning f(x) = x for every x in the first set. This mapping is clearly injective.

Here, we define a simple one-to-one mapping where each number in the first set maps to itself in the second set. Since neither endpoints (0 and 1) are included, any number chosen from the first set will find its place in the second set without conflict. It clearly maintains uniqueness as no two different numbers will map to the same number in the second set.

Imagine a line of children forming pairs with another line but without including the first and last ones. As long as every child chooses another child directly across from them, it’s easy to see there’s a perfect matching without confusion.

Consider the mapping g defined as g(x) = x/2. If x is from the second set, this will give output between 0 and 1, still excluding 0 and including 0.5 when x is 1.

In this mapping, we take each element from the second set and divide it by two. This effectively reduces every number, ensuring that it falls into the first set's realm as desired. This transformation will also preserve uniqueness, meaning that two different inputs will not result in the same output.

Consider a bakery where every muffin from the second batch is split in half to serve customers from the first group. Every half muffin remains distinct and tasty while still catering to the needs of customer A without confusing the categories of muffins.

Since we have demonstrated both mappings are injective, we can conclude that the sets have the same cardinality.

After establishing the injective mappings in both directions between the sets, we can confidently say they possess the same cardinality. This reinforces the idea that different sets, even if seemingly different, can have the same number of elements.

It’s like confirming that two different kinds of fruit baskets can hold the same number of apples, even if one has more large apples and the other more small ones. The actual quantity remains equal, demonstrating how appearances can sometimes be misleading.

--