22.1.1. Lecture -22

Hello everyone, welcome to this lecture on equivalence relations and partitions. Just to recap in the last lecture we introduced the notion of equivalence relation and equivalence classes. In this lecture, we will continue the discussion on equivalence relations and classes. And we will introduce the notion of partition of a set and we will see the relationship between equivalence classes and partitions.

This chunk serves as the introduction to the lecture. It highlights that the focus will be on equivalence relations and partitions, building on previous concepts. An equivalence relation is a way to group elements in a set based on some criteria, creating equivalence classes. A partition is a way to divide a set into non-overlapping subsets. Understanding the relationship between these two concepts is crucial in discrete mathematics.

Think about how students are grouped in a classroom based on their performance: you have high achievers, average students, and those needing extra help. Each group forms an equivalence class based on their grades. Now, if we consider all distinct groups or classes, it forms a partition of the entire student body.

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 C is defined as a collection of subsets that are pairwise disjoint and non-empty. Pairwise disjoint means that no two subsets share elements; they do not overlap. When you take the union of all these subsets, you should get back the original set C. For instance, if you have a set of fruits and you create subsets representing different types of fruits like apples, bananas, and oranges, each subset must have distinct fruits, and together, they should encompass all fruits from the original set.

Consider a library with a collection of books. If the books are categorized into sections like fiction, non-fiction, and reference, each section represents a partition of the library's collection. No book belongs to more than one section (disjoint), and all books should be included in one of these sections (covers the whole set).

So intuitively, say for example, you have the map of India you can say that the various states of India partition the entire country India into various subsets such that there is no intersection among the states here. So, in that sense, I am just trying to find out some subsets of the set C such that there should not be any overlap among those subsets and if I take the union of all those subsets I should get back the original set C, there should not be any element of C which is missing.

To qualify as a partition, the subsets created must meet certain criteria: they must be non-empty (each subset must contain at least one element), their union must equal the original set (no element should be left out), and they must be pairwise disjoint (no shared elements among subsets). This ensures that the entire set is completely and accurately represented by the partition.

Think of organizing a sports tournament. Each team plays in a category where they do not overlap with others (disjoint), every player must belong to a team (non-empty), and all players should be part of some team (union equals the whole set of players).

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 [...].

This chunk explains how equivalence relations are related to partitions. Given a set C, if an equivalence relation R is defined, the resulting equivalence classes formed by this relation will constitute a partition of the set C. This means that the way elements group together in equivalence classes reflects a partitioning of the overall set.

Imagine a party where guests are grouped by their favorite type of music. Each group (equivalence class) represents a different musical preference, such as rock, pop, or classical. Together, these groups divide the total guest list, demonstrating how preferences partition the group of attendees.

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. Then I can give you an equivalence relation R whose equivalence classes will be the subsets, which you have given me in the partition.

The text discusses the ability to reverse the relationship: if you start with a partition of a set, you can define an equivalence relation whose equivalence classes correspond to the subsets of that partition. This means that the two concepts of partitions and equivalence relations are intrinsically linked; each can be derived from the other.

Picture a classroom where students sit in different groups based on their projects. Each group is a partition of the entire class. Conversely, you can define a relation where students in the same project group are considered equivalent, forming an equivalence relation. Thus, you can switch back and forth between these two views.

So, in other words what we can show here is that the number of equivalence relations what we have established here actually is that the number of equivalence relations over C is exactly the same as the number of partitions of the set C.

This chunk concludes by stating that the number of distinct equivalence relations on a set is equal to the number of distinct ways to partition that set. This encapsulates the profound connection between these two concepts in set theory, showing that one can be studied through the lens of the other.

Consider a vending machine with different snacks. If you categorize snacks as chips, candy, and drinks, that’s a way of partitioning all possible snacks. Conversely, you could define relationships based on snack preferences, leading to equivalence classes. The number of ways to categorize snacks (partitions) corresponds exactly to how many distinct snack preferences (equivalence relations) exist.

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