 In many ways, mathematics is a little bit different from every other field of study, in that mathematics is not so much about things, but rather the relationship between things. And this is particularly evident in one of the most fundamental concepts of modern mathematics, which is the notion of a set. And we'll introduce several terms, and rather than trying to define what these things are, what we're going to do is we're going to define how these things relate to each other, and we'll define these terms relationally. So we might say, first of all, that a set consists of a collection of elements, which are said to belong to the set, and the set includes the elements. So here we indicate the relationship, set contains elements, elements belong to the set. What's a set? What's an element? Who knows? Who cares? The important thing is that we know how they relate to each other. And we do want to introduce some notation. This idea that an element belongs to a set, well, if X is an element of my set, I'm going to write X is an element of our set A, and this symbol here is kind of a fancy looking little E, and there's two common ways of describing sets. One, we can talk about what's called list notations, and this is kind of as what we would expect it to be. We're just going to list the elements of the set, and we're going to throw braces around those elements, and for readability, the elements of a set are going to be separated by commas. That's not actually necessary. It's just convenient because if we jumbled them all together, our set would be unreadable. The other way is we can write down what's called set builder notation, and this is going to be, we're going to describe a membership rule for who gets to belong to the set. For example, let's consider the set of whole numbers from one through five inclusive. So here's the verbal description of what our set is. If we want to write this in list notation, I want to list the elements and close to the set of braces. So there's my starting brace, and the things that are in this set, the numbers from one through five inclusive, so that's one, two, three, four, five, and I'll close off the set with that open brace, and so there's my set that includes the whole numbers from one through five inclusive. Now, I can also write this down in set builder notation, and what I want to do is I want to describe what you have to do in order to be in the set. And in some ways, the set builder notation is much more intuitive because it's actually just this. Our set builder notation uses our verbal description of the set. Now, we do want to formalize that a little bit because two people may describe the set in two different ways. We want to make sure that they're talking about the same set. So how we're going to do that is the following convention. Our set is going to consist of things where, and that's how we'll leave that vertical bar, and the membership requirement, what we are looking at. The thing is a whole number from one through five inclusive, and again, close our set with the braces. Now, in practical terms, it is more useful for us to be able to read set builder notation and go back to list notation, to try and go from list notation to set builder notation can be a little difficult because you have to figure out what it is that these things have in common that, importantly, nothing else in the entire universe has in common with these things. And so our set builder notation to here, that's important, to try and go from here to there, a little bit less important, far more challenging, and in some cases, far more useful in the broader scheme of things. But for our purposes, we want to be able to go set builder back to here, description to list to set builder. For example, let's consider our set S, and let's see this is things where the thing is a US state. And here's a common thing you'll want to be able to do, list three elements of our set S. And then, well, is it true that Guam is an element of S and how about New York City an element of S? Now let's take a look at that. So the elements of S are things that are US states. And so let's think about this. So we might list three states. How about New York, California, Canada? Oh, wait, no. This is not a geography class, but you should know Canada is not a US state. How about Utah? So there's three things that are in the set S. Now let's think about that. Is Guam in this set? Well, Guam is not a US state. So it does not meet the membership requirements of this set. So Guam in S is not true. So how can we write that? Well, we can write Guam element of S not using that slash. Likewise, New York City is not a US state. So New York City is not an element of the things that are US states.