22.5. Proof of Properties of Equivalence Relation

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 relationship between equivalence relation and partition of a set. So, imagine you are given a set C consisting of elements. Now what I can prove here is that if R is an equivalence relation over the set C and if the equivalence classes which I can form with respect to the relation R are constituted, then my claim here is that the equivalence classes constitutes a partition of the set C.

This chunk introduces the fundamental goal of the section: to prove that equivalence classes formed by an equivalence relation on a set result in a partition of that set. An equivalence relation groups elements based on a defined relation (like equality) and these groups (or classes) do not overlap and cover the entire set, satisfying partition requirements.

Think of a classroom where students are grouped by their favorite subjects. If you were to partition the students based on these subjects, each student belongs to one unique group, corresponding to their favorite subject, and together these groups make up the entire class. This mirrors how equivalence classes operate within a set.

So, just to recall, the definition of partition demands me to prove three properties, 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 second requirement from the partition is that the union of the various subsets should give me back the original set. So, my claim here is that if I take the union of all these equivalence classes, I will definitely get back my original set C. Third requirement from the partition was that the various subsets in the partition should be pairwise disjoint.

In this chunk, three key properties necessary for a set to be a partition are identified. First, every subset (equivalence class) must contain at least one element, which is guaranteed by the nature of equivalence relations. Second, all these subsets when combined (union) must cover the entire original set. Finally, the subsets must not overlap (be pairwise disjoint), which is a crucial aspect that ensures clarity in classification.

Consider a library that organizes books into genres. Each genre (like fiction, non-fiction) must have at least one book (non-empty), all the books in the library should fall under some genre (covering the whole set), and no book should belong to more than one genre (no overlap), mirroring the requirements of a functional partition.

So, in this specific case, I have to show that you take any two equivalence classes, they should be pairwise disjoint and that is easy because in the last lecture we proved that two equivalence classes are either same or they are disjoint. You cannot have a common element present in two different equivalence classes which automatically establishes that these subsets are pairwise disjoint.

This part asserts that any two equivalence classes cannot share elements, which means they must be disjoint. This property is fundamental in showing that when constructing a partition from equivalence classes, there will not be any overlap or ambiguity about which class an element belongs to.

Imagine sorting fruits into baskets by type (apples, oranges). An apple cannot be found in the orange basket and vice versa. Each fruit belongs to one type only, similar to how each element of a set belongs to a single equivalence class.

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 subsets which constitute a partition of the set C. Then I can give you an equivalence relation R whose equivalence classes will be the subsets which you have given me in the partition.

Here, the text begins to discuss the duality between partitions and equivalence relations. It states that not only do equivalence relations lead to partitions, but you can also start with a partition and create an equivalence relation from it, demonstrating the interconnectedness of these concepts.

Think of a school’s system where each student is categorized by interest groups (sports, arts). This organization serves as a partition. Now, if we define a relation where students in the same interest group are considered 'related', we’ve essentially formulated an equivalence relation based on the original partition.

Let us prove that a relation R that I am saying here will be reflexive, symmetric, and transitive. For proving that my relation R is reflexive, Symmetric and Transitive, I take any element i from the set C, and I must show that (i, i) is in R for reflexive property. Likewise, for symmetry, if (i, j) is in R, then (j, i) should also be in R, and for transitive property, if (i, j) and (j, k) are in R, then (i, k) should be in R.

This chunk focuses on proving whether the constructed relation R maintains the properties of an equivalence relation. These properties are essential for any relation to qualify as an equivalence relation, ensuring that elements are related appropriately based on the established criteria.

Consider a friendship relation in a network where if Alice is friends with Bob (reflexive), then it must also hold that Bob is friends with Alice (symmetric). If Alice is friends with Bob and Bob with Charlie, then Alice must also be friends with Charlie (transitive). This translates well to how we define relationships in mathematical equivalence.

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