21.6. Disjoint Equivalence Classes

Imagine R is an equivalence relation over some set A. And now consider an element a ∈ A. Then the equivalence class of a which is denoted by this notation you have the square bracket and within that you have the element a. So, the equivalence class of [a] = {b:(a,b) ∈ R}, consists of all the elements from the set A which are related to this element a as per the relation R.

In this chunk, we define what an equivalence class is in the context of set theory. An equivalence class groups together elements that are considered equivalent under a certain equivalence relation R. If you have a specific element 'a' from your set 'A', the equivalence class of 'a' (denoted [a]) includes all elements 'b' in 'A' that relate to 'a' through the relation R. Essentially, it categorizes elements based on how they relate to each other.

Think of a class of students where students are grouped by their study habits. If 'a' is a student who studies every night, then the equivalence class [a] would include all students who share that trait. You could say, 'All students who study nightly are in the same group,' meaning they are equivalent in the context of their study habits.

Formally, this equivalence class is a set it will be a subset of your set A. It will be having all the elements b ∈ A such that a is related to b. That is equivalence class of an element a. And now this equivalence class satisfies some very nice properties. The first trivial thing to check here is verify here is that you take the equivalence class of any element, it will be non-empty. There will be at least one element which is always guaranteed to be present in the equivalence class of any element a. And that element is the element a itself.

This section outlines the fundamental properties of equivalence classes. Firstly, we highlight that any equivalence class is a subset of the original set A. Importantly, every equivalence class must contain at least the element 'a' itself, confirming it can never be empty. This is due to the reflexive nature of equivalence relations which states that every element is related to itself.

Imagine you have a box of colored balls. You could have red, blue, and green balls. Each color represents an equivalence class. No matter which colored ball you pick, the class will always include that ball itself - you cannot have a class without it. So, if you pick a blue ball, you automatically belong to the blue equivalence class.

Let me demonstrate what exactly equivalence class looks like with an example. So, I consider this relation R over set of integers Z where an integer is related to integer b if a ≡ b (mod 3). So, what will be the equivalence class of 0? So, [0], so my a = 0 here, so equivalence class 0 will have all the elements b, empty all the integers b such that 0 is related to those integers b.

In this chunk, we provide a specific example of equivalence classes using modulo arithmetic. When we say two integers a and b are related if a is congruent to b modulo 3, we illustrate how the equivalence class [0] contains all integers that leave a remainder of 0 when divided by 3, such as ..., -6, -3, 0, 3, 6, ... . This example visually demonstrates how numbers group into classes based on their remainders when divided by the modulus.

Think of a monthly schedule where weeks are repeated every 7 days. If 'Day 0' is a Monday, then the equivalence class of Monday includes all Mondays for the entire year. Just as all those days belong to the same category of 'Monday', numbers that are multiples of 3 belong to the same equivalence class under the modulo relation.

So, what we are observing here is that even though we have equivalence class of every integer possible here. So, these are the various equivalence classes. And this is an infinite list. It turns out that if we closely look here we find that the two equivalence classes in this sequence are either same or they are completely disjoint.

In this section, we discuss a crucial characteristic of equivalence classes: they are either the same or completely distinct from one another, meaning they share no elements. This concept is key in set theory and underscores the importance of equivalence relations in categorizing elements of a set without overlap.

Imagine sorting people into groups based on their favorite colors. If one group loves blue and another loves yellow, there can be no one who is in both groups at the same time. Just like with equivalence classes, you either belong to one group or the other, highlighting the disjoint nature of these classifications.

More formally we can prove that if you are given any equivalence relation, any arbitrary equivalence relation over an arbitrary set then a is related to b iff they are equivalence classes are same and the equivalence classes as [a] = [b] if and only if [a] ∩ [b] ≠ ∅.

This chunk formalizes the relationship between elements and their equivalence classes. Specifically, it asserts that two elements a and b are considered related if and only if their equivalence classes are identical. If they are different, their equivalence classes will have no overlap at all—meaning the intersection of that classes is empty (∩ = ∅). This emphasizes the mutual exclusivity of equivalence classes.

Consider two distinct clubs—one for chess players and one for soccer players. There are no members who belong to both clubs, demonstrating how some distinctions are clear-cut. Similarly, students who belong to different equivalence classes, say based on different traits or characteristics, do not share any common membership.

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