 So good, let's talk about equivalence relations. I'm going to throw quite a few definitions at you in this little section. So let's just talk about equivalence relations, equivalence, equivalence relations. Now, we've looked at binary relations and we've looked at its properties. Now, if it has all three of those properties, if it is reflexive, reflexive, if it is symmetric, symmetric, and it is transitive, it has all of those three properties. We call it an equivalence relation. And there's something very special about equivalence relations. And I'll show you what that is. Before we do that, though, let's just get another definition. Let's have a look at something like parity. I just want to do that as an example. Let's look at parity. So if we just look at the set of integers, elements of the integers. So elements of the integers of the integers have parity if they are either even or odd. So 2, 4, 6, they're all even. So they have parity, 1, 3, 5, they're all odd. So they will have parity. So let's look at this relation. If we have our relation has the same parity, let's do that. Our relation is has same parity. That is our, that is our. And we just look at the set that we are looking at as the set of all integers minus 0. So we're just taking all the integers except for 0. So negative 1, negative 2, and 1, 2, 3, all of those. And we just look at as the same parity. So what are we going to be left with? Well, we're going to be left actually with two subsets, aren't we? Our first subset that we're going to have, subset 1, is going to be the set of all elements. Let's call them, what did I call them? In my example, I call them n, such that n is an element of the integers minus 0, and n is even. And n is even. That will be my first subset. And my second subset is going to be the set of all n, such that n is an element of the integers minus 0. And n is odd. Now look at those two. If my set that I started off with is the set of integers minus 0, if that is the set that I'm looking at. Look what happens here if I make the union of these two sets. Well, that is going to equal my set that I started with. And if I look at my intersection of these two, a sub 1 and a sub 2, that is going to be the empty set. That is going to be the empty set. And let's have a look at it. Is this thing reflexive? This relation of mine, is it reflexive? So for any element, say in this first one, in this first subset that I've created, I can have 2 comma 2 and 3 comma 3. For any element that I choose in this one, I can write this. So it definitely is that. And if I look, is it symmetric? Well, if 2 and 4 is an element of that, 4 and 2 is definitely an element of that. So it definitely is symmetric. Is it transitive? If 2 and 4 is in it, and let's make it 4 and 6 are in there, 4 and 6 are in there. And then most definitely 2 and 6 are also in there. And indeed, 2 and 6 would also be in there. So this is an equivalence. This is an equivalence relation. And the nice thing about an equivalence relation is it creates a partitioning. A partitioning of my set that I started off with. And we know it's a partitioning. If I take the union of all of these subsets that I create through this equivalence relation, I get back to the original set. And if I take the intersection of any two, any two partitions, I might have 1,000 partitions here. Partitioning up my original set in these, I might have 1,000. Any two of them, their intersection will be empty. So the union of all of them will be the original set. Any pair of them, if I take the intersection of any pair, that will be an empty set. So that is an equivalence relation that has an equivalence relation as a symmetric. We're just dealing with symmetric now, a symmetric relation that has all three of these properties with this unique, unique little trick to it that it has partitions, at least my original set.