22.1. Discrete Mathematics

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 a set into smaller subsets. These subsets must not overlap (called pairwise disjoint), and they must cover the entire original set without leaving any elements out. This means that when you combine (or take the union of) all these smaller subsets, you return to your original set C.

Think of a partition like dividing a pizza into slices. Each slice represents a non-overlapping portion of the entire pizza (the original set). If you put all the slices back together, you recreate the whole pizza without losing any cheese or toppings.

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 C_i, C_j then C_i ∩ C_j = ∅ and C_i ∪…∪C_m = C. So, one trivial partition of the set C is the set C itself.

For a collection of subsets to be a partition, they must satisfy a few specific rules: each subset must contain at least one element (no empty subsets), they cannot overlap at all (the intersection of any two subsets is empty), and if you combine all the subsets together, you must get back the original set. An example of the simplest partition is just considering the original set as a single subset without breaking it down at all.

Imagine sorting different colored marbles into jars. Each jar holds marbles of only one color, and at least one marble is in each jar. If one jar had no marbles, it would violate the rules. The union of all jars must result in all the marbles you had originally.

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...

An equivalence relation is a way of grouping elements based on a certain property that relates them. When you have an equivalence relation defined on a set, you can form equivalence classes, which are like partitions where every class is a subset containing elements related to each other. The relationship we are discussing is that every equivalence relation can create a partition of the set, and conversely, every partition can lead to an equivalence relation.

Think of a classroom where students are grouped based on their favorite subject. If math is one group, science another, and art another, each student's favorite subject relates them to their group, just like an equivalence relation. All students associated with math together form a partition, splitting the entire class based on their preferences.

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.

If you start with a partition of a set, you can construct an equivalence relation by saying two elements belong to the same relation if they are in the same subset of the partition. This means every subset from the partition defines its corresponding equivalence class, thus forming an equivalence relation.

Consider a library where books are divided into sections based on genres: fiction, non-fiction, and mystery. Each section is a partition; if two books are in the fiction section, they are related, as they belong to the same genre group, forming an equivalence relation based on genre.

So, let us formally prove this. Now, going to prove that a relation R that I am saying here to construct indeed will be reflexive, symmetric and transitive...

To show that the constructed relation R is indeed an equivalence relation, we must prove three properties: reflexivity (every element is related to itself), symmetry (if one element is related to another, then the second is also related to the first), and transitivity (if one element is related to a second, and that second is related to a third, then the first is related to the third). These properties help validate that our construction behaves as expected.

Imagine a relationship of 'friendship' among students. Every student is friends with themselves (reflexivity), if student A is friends with student B, then B is also friends with A (symmetry), and if A is friends with B and B is friends with C, then A is also friends with C (transitivity). This friendship relation mirrors the properties of an equivalence relation.

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 number of partitions of set C.

In conclusion, we’ve established a fundamental connection between equivalence relations and partitions. For every possible equivalence relation on a set, there exists a corresponding partition, and vice versa. This means that understanding and counting one gives insight into the other, highlighting the integral relationship between these concepts.

It's like analyzing a network of friendships. Any way of grouping friends based on who is friends with whom can lead to different groups (partitions), and each unique group dynamic can be viewed as a different 'friendship rule' or equivalence relation. Thus, the methods of counting these friendships (relations) and groups (partitions) will yield the same results.

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