22.3.1. Requirements for Partition

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 to divide the set into smaller groups where each group contains at least one element and no two groups share any elements. This means that every element of the original set must be included in one of these groups, and there should be no overlap between the groups. For example, if we have a set of students in a class, we can partition that set into different groups based on their favorite subjects, ensuring that each student is only in one group.

Imagine a pizza divided into slices. Each slice represents a non-empty subset, and together, all slices make up the whole pizza without any overlap. You can’t have one slice of pizza that overlaps with another slice; they must fit together perfectly to form a complete pizza.

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, their intersection is empty, i.e., no element should belong to both subsets. Additionally, the union of all subsets should equal the original set C.

For a collection of subsets to be a valid partition of a set C, there are three key conditions that must be met: (1) Each subset must have at least one element; (2) No two subsets can share any elements, meaning they are disjoint; and (3) When you combine (union) all the subsets, you must get the original set back without missing any elements. These conditions ensure that the partition accurately represents all parts of the original set.

Think of a school with different classrooms. Each classroom (subset) has students (elements), and no student belongs to more than one classroom at the same time. This ensures that every student is accounted for (the union of all classrooms makes up the entire student body), and there are no empty classrooms.

So, one trivial partition of the set C is the set C itself. I can imagine that C is partitioned into just one subset namely the entire set C or I can decide to partition C into exactly two halves or I can decide to partition C into three equal sets of equal sizes and so on.

A trivial partition refers to a case where the entire set is considered as a single subset. For instance, if you have a set of numbers, you can consider the partition that consists of just that one complete set without breaking it down further. Additionally, you can create different partitions by dividing the set into equal or uneven groups, depending on how you'd like to classify or group the elements.

Imagine a fruit basket containing apples, oranges, and bananas. Considering the entire basket as one partition (the trivial partition) means you are looking at all the fruits collectively. Alternatively, you could partition the fruits into different groups, such as one group for apples and another for oranges, based on their types.

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, 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, the equivalence classes form a partition of the set.

An equivalence relation is a special type of relation that groups elements based on their characteristics. When we create equivalence classes from these relations, we can represent these classes as a partition of the set because they meet the criteria we discussed earlier: they are non-empty, disjoint, and their union covers the entire set. This means every element belongs to exactly one equivalence class, which corresponds to the idea of partitioning.

Consider a group of people sorted by their favorite sports. Each sport can be viewed as an equivalence class, with all fans of that sport grouped together. Each group of sports fans (each class) does not overlap with others since a person can only be in one class based on their favorite. Together, all sports fans cover everyone in the group, demonstrating a perfect partition.

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, 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 whose equivalence classes will be the subsets, which you have given me in the partition.

This reverse proposition states that not only do equivalence classes form partitions, but given any partition, we can define an equivalence relation such that the equivalence classes correspond exactly to that partition. This means we can 'construct' equivalence relations based on existing groups or partitions.

Think of a collection of books sorted by genre: fiction, non-fiction, and science fiction. Each genre constitutes a partition of the book collection. You can define an equivalence relation where two books are equivalent if they belong to the same genre. Thus, the partition of books into genres and the equivalence relation based on genre are closely linked.

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