22.4.2.1. Construction of the Equivalence Relation

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 the set into non-overlapping pieces. Each of these pieces is called a subset, and they must fit together perfectly to recreate the original set, meaning that no elements are left out and no element appears in more than one subset. For example, if we think of a set as a collection of fruits, a partition could be separating them into different categories like apples, bananas, and oranges.

Think of a classroom where students are categorized based on their favorite sports. Each group (soccer, basketball, and tennis) represents a subset of the total class. Each student belongs only to one group, ensuring no overlaps. When we combine all the sports groups, we return to the original class of students, illustrating how subsets partition the set.

So, more formally the requirements here are the following. Each subset ≠ ∅; that means each subset should have at least one element. They should be pairwise disjoint. That means if I take any A,B then A ∩ B = ∅ and A ∪…∪A = C.

For a collection of subsets to be a valid partition, they must meet three key properties: each subset must contain at least one element (they cannot be empty), all subsets must not overlap with one another (they are pairwise disjoint), and together, the union of all these subsets must equal the original set.

Consider organizing a party with a guest list divided into categories (friends, family, coworkers). Each category has at least one person (not empty), no one belongs to more than one category (disjoint), and every invited guest fits into one of these categories without omitting anyone from the total guest list (union gives back the original set).

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 the relationship between equivalence relation and partition of a set.

This chunk of content introduces how equivalence relations—specific kinds of relationships defined on a set—connect directly to the concept of partitions. Every equivalence relation creates equivalence classes which can be thought of as divisions of the set, just like a partition. Essentially, when you have an equivalence relation, it provides a natural way of partitioning the set.

Imagine a school where students are grouped based on their grades (A, B, C). The grades form equivalence classes where all students in class A are equivalent (same grade) and together they partition the entire student body. Each grade level is disjoint, ensuring a clear and organized system for understanding student performance.

The first property is that each of this subset 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.

The first requirement of a partition is that none of the subsets can be empty. By the nature of equivalence classes, each class contains at least one member, which satisfies this requirement effortlessly. This ensures that every category made by the partition is useful and relevant.

Returning to the earlier example of guests at a party, if one category (like friends) had no one attending, the category wouldn't serve its purpose. In the same way, equivalence classes derived from relations should have at least one representative to maintain the integrity of the categorization.

The second requirement from the partition is that the union of the various subsets should give me back the original set. The third requirement from the partition was that the various subsets in the partition should be pairwise disjoint.

These two requirements assert that when you combine all of the subsets that make up your partition, you need to get back your original set without any elements missing (union) and ensure that no element is present in more than one subset (pairwise disjoint). This fits the careful design of how you create these subsets.

If the students mentioned earlier are split into sports categories, combining members from all categories leads back to the total number of students originally invited. Additionally, no student can be both a basketball player and a soccer player, thus keeping the groups distinct and organized.

So, we have proved here that you give me any equivalence relation and if I take the equivalence classes that I can form with respect to that relation R that collection of equivalence classes will constitute a partition of my original set.

Here, the argument is made that the equivalence classes created by any equivalence relation inherently satisfy the properties needed for a partition, meaning they will always form a valid partition of the set. This conclusion ties the concepts together, showing that equivalence classes are indeed a direct method of partitioning a set.

Using the classroom sport categories again, if you define students based on their preferences, each category formed (basketball, soccer) shows that every student is accounted for and separated correctly, confirming they can be partitioned efficiently.

Now, I can prove the property in the reverse direction as well. What do I mean by that? I claim here that you give me any partition of a set C, say you give me a collection of m subsets which constitute a partition of the set C.

The segment discusses how the relationship is reversible: if you start with a partition, you can construct a corresponding equivalence relation that categorizes elements in a way that results in the original partitions as equivalence classes. This bidirectional property highlights the equivalence between the two concepts.

If we think of a bookshelf where books are organized by genres (mystery, science fiction, biography), this grouping can be viewed as a partition. If we now create a system (equivalence relation) that links all mystery books together, we've shown how one concept informs the other effectively.

So, just to demonstrate my point, imagine my set C = {A, B, C, 4, 5, 6} and a partition of this set is given to you. So, I am given 3 subsets, C1 = {A, B, C}, C2 = {4, 5}, C3 = {6}...

The example illustrates how to construct an equivalence relation from a given partition by iterating through each subset to add pairs of elements into the relation. Each subset’s members are related to each other based on their grouping, forming a clear structure of equivalences.

If you think of a family reunion where family members are grouped by their last names (Smiths, Johnsons), each family connection forms a relationship where every member of a family is connected. As we form this relationship, we not only love the family, but we also see how these ties fit perfectly into one large family picture.

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