22.4.1. Equivalence Classes as Partitions

So, let us start with the definition of a partition of a set. So, imagine you are given a set C which may be finite or it may be infinite. Now, what is the partition of this set C? The partition here is basically a collection of pairwise disjoint, non-empty subsets say m subsets of C which should be pairwise disjoint such that if you take their union, you should get back the original set C.

A partition of a set is a way of dividing that set into smaller subsets in such a way that these subsets do not overlap (they are disjoint) and when combined (their union), they recreate the original set. If you think of a set as a group of items, partitioning it means forming different categories or groups where every item belongs to exactly one group, and all items are covered without leaving any item out.

Consider a box of assorted fruits: apples, bananas, and oranges. If we say our set C is the box of fruits, we can create a partition by putting all apples in one basket, all bananas in another, and all oranges in a third basket. Each basket is non-empty, no basket contains the same types of fruit (pairwise disjoint), and if we gather all the fruits from the baskets, we get back the original box of assorted fruits.

So, more formally the requirements here are the following. Each subset should have at least one element. They should be pairwise disjoint. That means if I take any two subsets, then their intersection is empty. And the union of all these subsets should give back the original set C.

For a collection of subsets to qualify as a partition, three criteria must be met: each subset must contain at least one element, no two subsets can share members (they are pairwise disjoint), and together all subsets must recreate the original set when combined. These criteria ensure that every element in the original set is accounted for and that there is no overlap among subsets.

Imagine a classroom where students are assigned to different groups for a project. Each group must have at least one student (non-empty), no student can be in more than one group simultaneously (pairwise disjoint), and when you combine all the groups together, all students are represented (the union gives back the original group of students in the classroom).

What we now want to establish here is a very interesting relationship between the equivalence from an equivalence relation to the partition of a set. So, we want to establish a relationship between equivalence relation and partition of a set. So, imagine you are given a set C consisting of n elements. If R is an equivalence relation over the set C and the equivalence classes formed with respect to the relation R are C1, C2,..., Cm, then these equivalence classes constitute a partition of the set C.

An equivalence relation on a set establishes a way to group elements that are considered equivalent. The result of this grouping is called an equivalence class. The key point is that the collection of these equivalence classes completely partitions the original set. This means each element of the set is included in one and only one equivalence class.

Think of a group of people based on their seating arrangement in a theater. Each seating section (like left, middle, and right) can be viewed as an equivalence class where people in the same section are considered 'equivalent' regarding their location. When you gather all sections together (these equivalence classes), you recover the entire audience (the original set).

The first property is that each of these equivalence classes should be non-empty. And that is trivial because I know that each of these equivalence classes is non-empty because each of these equivalence classes is bound to have at least one element, an element related to itself. The second requirement is that the union of the various subsets should give back the original set. If I take the union of all these equivalence classes, I will definitely get back my original set C.

Every equivalence class must contain at least one element. This is guaranteed by the nature of equivalence relations since every element must be related to itself. Additionally, when you collect all equivalence classes together (take their union), they cover the whole original set without missing anything. This highlights how an equivalence relation divides the original set into distinct subsets, ensuring that all elements are included.

Consider a library organized by genre: each genre (e.g., fiction, non-fiction, mystery) represents an equivalence class. Every genre has at least one book (non-empty), and combining books from all genres covers the entire library collection (the union gives back the entire set of books).

Now, I can prove the property in the reverse direction as well. I claim here that you give me any partition of a set C. Then I can give you an equivalence relation R whose equivalence classes will be the subsets provided in the partition.

You can also start with a partition and create an equivalence relation. This relation defines equivalence among elements based on which subset they belong to in the partition. Each member in a subset is equivalent to every other member in that same subset, establishing an equivalence relation from the given partition.

Using our earlier example of genres in a library: if we define an equivalence relation based on the genres, all books in a genre are equivalent to each other. For instance, all mystery novels belong to the same equivalence class. Thus, starting from the genres (partition), we create an equivalence relation among all the books.

So, that shows a very nice relationship and a nice property between the equivalence classes and the partition. In short, the number of equivalence relations over C is exactly the same as the number of partitions of set C.

This means that for every possible way to partition a set, there is a corresponding equivalence relation, and vice versa. Thus, the two concepts are fundamentally linked, with the nature of how elements can be grouped or categorized reflecting in both partitions and equivalence relations.

Imagine a city’s public transportation system, where each bus route represents a part of the city's travel options (a partition). Each rider can choose multiple routes based on their destination, representing various equivalence classes among riders. Just as you can rearrange routes and riders, you can reframe equivalence relations or partitions, yet the total system remains intact.

--