 What can we do with an equivalence relation? We can define an equivalence class. Suppose squiggle is an equivalence relation on the elements of some set f. For any a in our set, the equivalence class of a is the set of all things in our set, f, where a squiggle x, and we say that a is the class representative. It's important to understand that equivalence classes are actually sets, so everything we know about how sets act and operate is going to apply to equivalence classes. However, sometimes we use notation like square bracket a to indicate these equivalence classes. Now since equivalence classes are sets, we might consider some of the questions we could ask about sets, and the standard questions you should always ask about sets are the following. Suppose a and b are sets. Always ask the following questions. Are they non-empty, and what's a intersect b? Well, let's consider those two questions. Suppose we have an equivalence class. Let's prove or disprove that our equivalence class is non-empty. So definitions are the whole of mathematics. All else is commentary. We're talking about equivalence classes. So in order for there to be an equivalence class, there has to be some equivalence relation. We'll call it squiggle. So our equivalence class consists of all x, where a squiggle x. Definitions are the whole of mathematics. All else is commentary. Since squiggle is an equivalence relation, we should remember what an equivalence relation is. So since it is an equivalence relation, we know it is reflexive, and so we know that a squiggle a and that means a itself must be in the equivalence class. In other words, every class includes its class representative. So we know our equivalence classes are non-empty. The other thing we should always ask about a set is suppose I have two of them. What's their intersection? And for this, it's useful to keep in mind. You can assume anything you want as long as you make it explicit. So suppose I have x in our intersection. So definitions are the whole of mathematics. All else is commentary. Remember that if x is in the intersection of two sets, then x has to be in each set individually. So I know that x is in the equivalence class of A, and x is also in the equivalence class of B. Definitions are the whole of mathematics. All else is commentary. Since it's in the equivalence class, we know that A squiggle x and also B squiggle x. Definitions are the whole of mathematics. All else is commentary. Because squiggle is an equivalence relation, we know that it's reflexive, symmetric, and transitive. And because we have three terms here, A, x, and B, transitive seems relevant, except we're not quite in the right form for transitive. Since in transitive, the second term of one relation has to be the first term of the other. But our relation is symmetric, and since B squiggle x, we can reverse it x squiggle B. Then, because we have an equivalence relation, we have transitivity, and so A squiggle B. And since A squiggle B, then B is in the equivalence class of A. At this point, I'm going to introduce something new that may cause my mathematician card to be ripped up. But here goes. If you're an approved course, you're probably pretty serious about studying mathematics. And one of the important things you want to develop is a sense of when something interesting is going on. And here's a useful guide. There are only three numbers, 0, 1, and infinity. Well, that's not really true, but it conveys an important idea. Either something doesn't exist, 0, it's unique, 1, or it occurs infinitely often. Another way to put it, 2 is an odd number. And in this particular case, it's very peculiar that if we have one thing in the intersection, we actually have a second thing that's in both sets, since remember B is in its own equivalence class. And because of that, we might consider the following. Suppose I have something else that's in B. Again, we can assume anything we want as long as we make it explicit. So again, definitions are the whole of mathematics, all else is commentary, because y is in the equivalence class, then we know B squiggle y. We already know that A squiggle B, because squiggle is an equivalence relation, then symmetry tells us y squiggle B and B squiggle A. And wait, get back here. Because squiggle is an equivalence relation, we have transitivity. So y squiggle A, and again, symmetry, so A squiggle y. So y is also in A. And if we put this together, if there is anything in the intersection, then anything else in B is also going to be in A. This means the entire equivalence class is in A. And it's useful at this point to take advantage of symmetry. Since everything we've said also applies, if we swap the roles of the equivalence classes of A and B, we can show the equivalence class of A is a subset of the equivalence class of B. And so if A is a subset of B and B is a subset of A, then the two sets must in fact be equal. And this leads us to the following theorem. If the intersection of two equivalence classes is not empty, if there is something in that intersection, then the two equivalence classes are identical. And finally, a useful thing to always keep in mind, the contrapositive has the same truth value. So if two equivalence classes are not the same, then their intersection must be the empty set.