 So, in many ways, modern mathematics is really the mathematics of sets. So, we'll introduce some basic terminology, and as an important idea in modern mathematics, it's actually impossible to define everything. What we have to do is we have to define things relationally. We have to say how ideas are related to each other. So, we're going to drop a whole bunch of terms here. So, we'll begin with the idea of a set, and this is a collection of things that we call elements, and we say that these elements belong to the set, or if we want to phrase it differently, we can say that the set includes the elements. Now, a little bit of notation there, because we want to be able to talk about an element that belongs to a set, we'll introduce some notation. If x is an element of our set A, we'll write x is an element of A, where we use this sort of funny-looking E to indicate element of. Now, there's two ways we can describe sets. Again, the idea is that since the set consists of elements, then we can identify what the elements are, and that will describe our set. There's actually two ways that we do this. The first and more intuitive is what's called list notation, and what we're going to do is we're just going to list the elements of the set, and we're going to enclose them in a pair of braces, the curly brackets. The other option is to use what's called set-builder notation, and what we're going to do here is we're going to describe a membership rule for the set. What an object has to do in order to belong to the set. For example, we might try to describe the set of whole numbers from one through five inclusive. So again, the list notation is a little bit more intuitive. What I'm just going to do is I'm going to list the things in the set. I'm going to enclose them with a set of curly braces. So there's my opener. The elements are going to be one, two, three, four, and five, and I'm done. So I do want to enclose the entire set with the n-brace there. And we can also try and describe the set in set-builder notation. So as before, we're going to open the set notation with a set of curly braces, and this time we're going to describe what something has to do in order to belong to the set. So how we write set-builder notation, I'm going to use a placeholder. I'll call that x, and the membership rule is that x gets to belong to the set if x is a whole number from one through five inclusive. And again, I do want to include that n-brace. Notice that in this particular case, our set-builder notation is not particularly compact, whereas in list notation, we have this nice compact set. However, in general, our set-builder notation is going to be more useful for describing sets with more than just a few elements. For example, let's take a look at a couple more sets. Let s be the set of things. So we could tell it's a set because of the braces. And it's things where what we're talking about is a U.S. state. And a useful thing to be able to do is to start off by listing three of the elements in s. That'll give us a handle on what s looks like. And then let's see if we can answer a few questions. Is Guam an element of s? Is New York City an element of s? So let's see. Well, let's try and find a few things that belong to s. The elements of s are things that are U.S. state. So let's list a couple of states. So this is not a geography course, but, well, New York's probably a good place to start. California, Canada. Oh, wait. Canada isn't a U.S. state. Utah. How about those three? So there's three things that are U.S. states. On the other hand, Guam is not a U.S. state. So Guam being an element of s is not actually a true statement. And we can write this in set notation by indicating that Guam element of s not. And by a similar argument, New York City is not a U.S. state. So New York City also is not an element of s. Okay, what else can we do? Well, suppose I have two sets, a and b. I can talk about a relationship between them. And the first relationship we might consider is when the two sets are equal. And we define them as follows. Sets a and b are equal if every element of the one set is also an element of the other set, and vice versa. This means that the sets contain exactly the same elements. And our notation for that is pretty much what you'd expect. We can write a is equal to b. The second possibility is that b might be a subset of a if every element of b is also an element of a. Now, we make a distinction here and we talk about proper subsets if we omit at least one of the elements of a. And in either case, we can write b is a subset of a, or sometimes we reverse that and write a. This symbol reversed from that symbol a contains b. All right, let's take a look at a couple of possibilities here. So again, let's take a few sets here. Let s be things where what we're talking about is the U.S. state. And another set where what we're talking about is the state east of the Mississippi River and being a bunch of cities in the northeastern part of the United States. And we want to prove or disprove a couple of subset relationships. Now, one way we want to examine this is whether or not this subset relationship holds. We can see if it satisfies the definition of being a subset. So for reference, we'll put that down. b is a subset of a if every element of b is an element of a. Now, we'll compare our definition of b subset of a to what we want to prove or disprove. And if we make that comparison, we see that b is w. We see that a is s. And so I can make that change. And so w is a subset of s if every element of w is also an element of s. Now, to complete the proof or disprove as the case may be, we want to determine whether or not this is actually true. Is it in fact true that every element of w is also an element of s? So let's see if we can relate our observation to our definition. w is all the states east of the Mississippi River and s is all the U.S. states. Well, everything that's a U.S. state east of the Mississippi River is actually going to be a state. So everything in w is in fact in s. And so we can relate our observation to our definition. Maybe we'll write it something like this. Since every element of w is a U.S. state, then every element of w is also an element of s. So w is a subset of s. Now, here's an important thing. When you're writing up a problem like this, you do want to make sure that you include everything that is in green. So what about the second question? n is a subset of s. Well, we want to check to see if it satisfies the definition, which is going to require that everything in n be also an element of s. Now, here's where understanding what the set consists of is actually kind of useful. If I were to try and list some of the things in n, well, it's a city in the northeastern part of the United States. So these are things like New York City, Boston, Rentham, Massachusetts, a whole bunch of other places. And not any of these are states. None of those cities are states. So we might make the observation that the elements of n are cities. They're not states. They can't actually be in s. And so we want to relate our observation to our definition to give the conclusion. The elements of n are cities, which are not elements of s. So it is not possible for n to be a subset. And again, the essential part of the answer is the portion that's shaded in green. Notice that I did not shade this section here. That's just a little bit of notation. It's something we can write n is not a subset.