 In this video, we're gonna talk about the related notion of a partition to that of an equivalence relationship. It turns out that a partition is just a different way of defining the same thing. We're gonna show that a partition is equivalent to an equivalence relationship. Ha ha, very funny, I get it. But a partition, what a partition is is the following thing. So a equivalence relationship was defined based upon a relation. A partition we're gonna define using directed from sets. So let's say we have a set in play here. This could be a finite set, it could be an infinite set, doesn't matter. We have a set X and we define a partition on X in the following way. We have a collection of non-empty sets, X1, X2, X3, up to however many there are, right? So we have some, these are gonna be subsets of X here. So each of these values, X sub K, will be a subset of X. They're subsets and they're themselves are non-empty. So every set contains something. But then the important kicker here is that one, these sets are disjoint. That is, if you take the intersection of any two sets, you always get something that's empty. There is no element that belongs to both XI and XJ. And then lastly, if we take the union of all of these sets, all of these subsets, you get the whole set itself. So a quick example of that would be something like the following. Take X to be the set, just one, two, three, four, five. Just something very, very simple here. And as a partition, we could then take as the following. We're gonna have the first set just contain one. The second set will contain two and three. And then the last set will contain four and five. Just a very simple example. But this is an example of a partition. So because the union of all of the subsets has to equal the whole set, this means that every element belongs to at least one cell. So one is right here, two and three are right here, four and five are right there. Everyone belongs to something. If we take the union of all of these cells, we will get everything inside of X. But then also, if we take the intersection of any two cells, we're going to get that they're empty. There's nothing that belongs to the first two cells. There's nothing that belongs to the first and third cell together. And then there's nothing that's common to both of those two cells. So this is in fact, a partition. Now it turns out that one, if one has a partition in hand, you can actually define an equivalence relationship. We can define a relationship on the set X cross X by the following rule. That we say that X is equivalent to Y if they share the same cell in P, the partition. So if two elements are in the same class inside of the partition, then we could say those things are related to each other. And I claim that this relationship is an equivalence relationship. And so in fact, when you have any equivalence relationship, you have the equivalence classes. I claim that the equivalence classes form a partition. And if you have any partition, you can form an equivalence relationship by saying two things are related if they're in the same class. And so that's actually that statement right there is the statement of theorem 1.2.35 here, given an equivalence relationship twiddle on a set X. The equivalence class on X forms a partition in the way we just described. And conversely, if we have a partition, we can construct an equivalence relationship saying things are equivalent if they're in the same class. So this is an if and only if statement here. How does one then prove that? Well, we have to prove both directions. So let's first start off with an equivalence relationship. Say that twiddle is an equivalence relationship on X. And so we're then gonna say that the equivalence class is form a partition. To form a partition, we have to show that the sets are non-empty, their intersections are empty, but their union is the whole set. So if you take an equivalence class, like take the class that contains X, it's non-empty because X belongs to it by the reflexive property. So by the reflexive property, these cells are non-empty. Why is the union all of the set, right? If you take the union of every equivalence class, if you take the union of all equivalence classes, that gets the whole set because if you take an arbitrary element inside of X, it's contained inside of its equivalence class. And so the union of all equivalence classes will give X. So we got the union, the non-empty union there, but what about intersections? If you take the intersection of two distinct equivalence classes, these have to be different classes because of course, if you intersect a class with itself, you'll just get back the class again. If you intersect two equivalence classes, why is this empty? Well, let's assume for the sake of contradiction that it is non-empty. Let's say there is something that belongs, some element Z that belongs to the equivalence class of X and the equivalence class of Y. Well, that would mean that, so if Z belongs to the class of X, that means that Z is equivalent to X. And if Z belongs to the equivalence class of Y, that means that Z is equivalent to Y. Well, by the symmetric property, if Z is equivalent to X, that means X is equivalent to Z, right? So we've now used the reflexive property and we've used the symmetric property. No, notice what's going on there. And so then if we put this together, X is equivalent to Z and Z is equivalent to Y, then by the transitive property, notice how we've used all three axioms now, by the transitive property, X is related to Y. And that means that X is then an element of the equivalence class of Y. That is, X is a subset of Y. And then reversing the argument, you're gonna get that Y, the equivalence class of Y is a subset of the equivalence class of X. Basically here, you're forcing that X and Y are the same equivalence relationship. So the only way that two equivalence classes, their section is not empty is if they're the same class. But for a partition, we only require that their intersections be empty when they're different. And so we basically get a contradiction there, that the intersection between two classes has to be empty. This proves the first direction that if you have an equivalence relationship, then the set of equivalence classes forms a partition for the set. Let's go the other way around. Let's assume we have a partition on the set X. So partition means there's a bunch of cells inside of P. These are non-empty subsets of X. We know that the union of all of these cells gives us X. And the intersection of distinct cells gives us the empty set. So let's define a relationship using P by saying that X and Y are related to each other if X and Y are contained inside the same partition class. Why is that going to be an equivalence relationship? Well, and then we're gonna say that X box there is the class that contains X. Well, okay, so great. So X is related to Y means that X is inside of the class Y. That is, well, notice here that X is related to Y exactly when these two things are equal to each other. And we're gonna get an equivalence relationship from that. You can show that X is related to X because X is inside of X box there. You're gonna get the symmetric property because equality is symmetric. You're gonna get that this thing is transitive because equality is transitive. This is the thing is when we did examples of equivalence relationships, you often notice that it was related to equations and somehow, right? That we said these two things were equivalent if there was some related equation and the equality properties prove the equivalence properties, the same thing's happening right here. So a partition always forms an equivalence relationship in vice versa. So two equivalence classes on an equivalence relationship are either disjoint or equal and important consequence of this statement here. So the one thing I want you to get from this video is that equivalence relationships and partitions are one in the same thing. And so if we define a partition, we have an equivalence relationship. If we have an equivalence relationship, we have a partition. The two things are inseparable.