 Another useful idea in mathematics is that of a partition. And so we define the following. Given a set S, a partition of S is a collection of subsets S i, where every x in our set is in sum of the S i, none of the S i's are the empty set, and if I take two distinct subsets, in other words, S i not equal to S j, then their intersection is the empty set. And we refer to the subsets as blocks or cells or parts of the partition. And you might notice there's three different words we could use for the same thing, and we won't always be consistent. We'll talk a little bit more about that in a moment. What does this have to do with equivalence relations? Well, if we have an equivalence relation, then for any x in our set, we've shown the equivalence class of x is not the empty set. There is at least one element in that set, and the intersection of two equivalence classes is the empty set, whenever the two equivalence classes are not actually the same class. Consequently, let squiggle be an equivalence relation on our set, the equivalence classes form a partition of our set. Now let's introduce a useful strategy in mathematics. Always ask, can you go backwards? Now the reason this is useful is that because mathematics is so useful that different people will invent different mathematics for their own purposes. But if you can go backwards and forwards between two concepts, then the two concepts are really different ways of describing the same thing. And in fact we can go a little further. The more words we have for describing the same thing, the more important it is. And that's because if something is important, different people will think of it and they'll give it their own names. So in fact when we define partition, these subsets were either called blocks or cells or parts. And the fact that we use different terms is indicative of the importance of this concept. In this case, an equivalence relation squiggle gives us a partition. Could a partition give us an equivalence relation? So let's take a partition of some set and let's define a relation and the obvious relation we might try to define here is that x, squiggle, y, if, x and y are in the same block. Is this an equivalence relation? So definitions are the whole of mathematics. All else is commentary. We'll pull in our definition of equivalence relation and we see we have to check to see if squiggle is reflexive, symmetric, and transitive. We also have a definition for squiggle and because it's a definition, it gives us two conditionals. If x, squiggle, y, then x and y are in the same block and also if x, y are in the same block, then x, squiggle, y. So we'd like to say that x, squiggle, x. So if I take some x in one of these blocks, our definition says we can conclude x, squiggle, y, if x and y are in the same block. Well, x is in xi. And since x and x are in the same block, x, squiggle, x. We need to show symmetry so we can always assume the antecedent of a conditional. So suppose x is related to y. Our definition says that if x is related to y, then x and y are in the same block. But since the order that we list them doesn't actually matter, so y and x are both in xi. Definitions are the whole of mathematics. All else is commentary. If they're both in the same block, then y is related to x. Finally, we need to check transitivity. So again, we can always assume the antecedent of a conditional. So suppose x, squiggle, y and y, squiggle, z. We can set that in our destination, x, squiggle, z. If x, squiggle, y, then we know that x and y are in the same block, si. And y, z are both in sj for some j. And here's an important thing to notice. We change the index because while we know that both y and z are in the same block, we are not going to commit ourselves to being in the same block as x and y. Don't commit until you're sure. So we'd like to conclude that x is related to z. And that means we have to show that z is in si. Well, how can we do that when we know nothing about what si or sj look like? Definitions are the whole of mathematics. All else is commentary. What we do know is that the si's form a partition, and we have our definition of a partition. And here we see that if si is not equal to sj, then their intersection is the empty set. Well, that's not what we have. But remember, the contrapositive of a conditional is logically equivalent to the conditional. And here's a conditional, and its contrapositive would be, if the intersection is not empty, then the two parts have to be the same thing. And that's what we have, because our intersection contains y, then our two parts have to be the same subset. And that means z is in si, and so x is related to z. And what this means is that our relationship is actually an equivalence relation. And so every partition of s induces an equivalence relation. And let's summarize this. Suppose s is a set. Every equivalence relation produces a partition, and every partition produces an equivalence relation. This means that if we have an equivalence relation, we have a partition, and if we have a partition, we have an equivalence relation. In some communities, this is known as a twofer. If we have either, we have both. And what that means is that we can work with either one.