 Hello, and welcome to the screencast on equivalence classes. So what is an equivalence class? Let's motivate this idea with an example. In a previous video, we looked at the relation on the set W of all words in the English language defined by R is the set of all points W1, W2, such that W1 rhymes with W2. So we saw in the last set of videos that R here is an equivalence relation. It's a relation that is reflexive, symmetric, and transitive. And that's important for what follows. Let's just pick a word out of the English language, say the word book. And let's suppose we're writing a poem or a song, and the first line of this poem or song ends in the word book. Now, we might want to know what are my choices for how to end the next line. So assuming that I want lines to rhyme, that is. Let's list some things we could put. We could use look, nook, took. If I wanted something with more than one syllable, I could say mistook or overlook, or I could even just use the word book itself, although that would be kind of a boring choice for a poem. There are probably more words than just these that rhyme with book. If I were going to express this entire list as a set, we might say that our choices constitute the set, the set of all W in capital W, such that W rhymes with book. That would be correct, but a more economical way to say it, since we have an equivalence relation in front of us, is to say it's the set of all W in W, such that the pair of book, comma W, belongs to my relation R, the rhyming relation. So this set we're referring to is just the set of words to which book is related under the rhyming relation R. Now, before we move to the main idea here, here's a quick concept check. Look at these two sets A and B, and the only difference here is the ordering of the pairs here in the sets. Then which of the following is true? Is A equal to B, or is A not equal to B? So think about your answer and especially think about why your answer is correct. So the answer here is that these two sets are actually the same, and the reason they're the same is because R, the rhyming relation, is an equivalence relation, and in particular it's a symmetric relation. So we know that the ordering of the pairs doesn't matter if we're relating to objects. So if book, comma W is in the relation, then W book is also in the relation. It doesn't matter which order we think of these words in, they'll still rhyme. Here's another similar example. Another important equivalence relation on the set of integers was the congruence mod N relation. Specifically, let's choose N equal to 5 and recall that we had the relation tilde on the set of integers by saying that A was related to B, A tilde B, if and only if A is congruent to B mod 5. We saw that was an equivalence relation. In fact, we could change the 5 to any natural number we wish, and it would still be an equivalence relation. Then what is in the set of all x in the integers such that 3 tilde x? Well, what does this even mean? This consists of all the integers that are equivalent to 3. That is all the integers that are congruent to 3 mod 5. Well, what are some of those? Well, 3 is certainly congruent to itself mod 5 because 5 divides 3 minus 3. 3 minus 3 of course is 0. 3 is also congruent to 8 mod 5, and also 13, and 8, and 23, and so on. Also 3 is equivalent to negative 2, negative 7, negative 12, and so on. If we wrote out this entire set in roster notation, we'd have this set. Now the pattern is clear from these points what's actually in the set. All of these are separated by 5, and it contains the number 3. So both examples we've seen here involve taking an equivalence relation on a set and an element out of that set and asking what is the set of all objects that are equivalent to the element that I chose? What's the set of all words that rhyme with book? What's the set of all integers that are congruent to 3 mod 5? The general concept here is what's known as an equivalence class. We define this as follows. We're going to let tilde be an equivalence relation on a non-empty set A. For every element little a in the set A, the equivalence class of little a determined by tilde is a subset of capital A. We denote this square bracket A, consisting of all the elements of A that are equivalent to little a. In other words, the equivalence class of A is equal to the set of all X such that X is equivalent to A. So for example, if tilde is the rhyming relation on the set of words in the English language, then the equivalence class of book is just the set that we saw earlier. It's a set of all words that rhyme with book. And if tilde is the equivalence relation given by congruence mod 5 on the set of integers, then the equivalence class of 3 is the set that we saw earlier, the set of all integers that are congruent to 5 mod 3. That's what equivalence means in that context. In each case, the equivalence class square bracket X is the set of all points Y to which X is equivalent under the equivalence relation in under study. So let's end this video off with a concept check. In a previous video, we introduced the relation on the set of real numbers by saying X and Y are equivalent if and only if they have the same decimal expansion. We argue that this is actually an equivalence relation. So which of the following numbers is in the equivalence class of 4.10? Select all the ones that apply and come back when you're ready. So the answer here is A, C, and D. These points are in the equivalence class of 4.10 because they have the same decimal expansion as 4.10. That of course is the same thing as 4.1. And having the same decimal expansion means that they are equivalent to 4.1. So in other words, 4.1 is equivalent to 2.1. 4.1 is also equivalent to 0.1. And pretty clearly, 4.1 is equivalent to itself. But 4.1 is not equivalent to 4.5 because the decimal expansions are different. And the same is true for 1.4. So that's the concept of an equivalence class. In the next video, we're going to take a look at the properties that equivalence classes have. So stay tuned.