 All right, let's take a look at problem 1.9. Before we do that, let's talk about our standard random mathematical aphorism. And today's choice will be a reminder of how to fail a math class. And one of the ways that we have a failing a math class is to look for how similar examples are solved. When you're confronted with a new question, you look for similar examples and see how they're solved. And this is a guaranteed way to fail a math class because you will never ever be shown how to solve every variation of a problem. If you try to look for examples of how something is solved, you might be really lucky and find one, but it is more likely that you will get a variation of a problem that you have never seen solved before. So it's important to understand the reasoning behind the steps, but it isn't generally worth looking for similar examples. So let's take a look at this problem to find a one-to-one correspondence between two sets and set the lea-de-de-de-de. All right, so this is problem 1.9. And so let's take this problem apart. Our first question here is to explain how you could match an element of w with a unique element of c and then indicate why the matching is unique. And note that that means that this question here actually has two parts. In fact, the same is true for part b. Explain how you could do something and then say something about why the matching is unique. So both of these questions have two distinct components to them. So let's take them apart. All right, so let's take a look at this first question, first part of the question. Explain how. So here's an important idea because this is asking us how to do something that our answer to the first part of the question is going to be the description of a process. Here is what I'm going to do. And so what I want to do is I want to match an element in w to a unique element in c. And I'll worry about the uniqueness as the second part of the question. So I want to take something in w and I want to explain how I'm going to form that matching. Well, in order to do that, let's consider this w consists of the set of US states. c is the set of capitals of US states. And so what I want to do is I want to take something that is a state and somehow match it to something that is a capital. So let's see how I would do that. What is the thing that I would do? So, well, maybe we can start out this way. Paper is cheap. It doesn't hurt to write something. So what do I want to do? I want to match an element of w. Well, maybe I'll match. Well, w is the set of states. So I need to match a state to a unique element of c. Well, c is a capital of the US state. So I want to make sure that that matching is to a unique element. So once I have a state, which capital should I match it to? And if I can explain which capital, that answers the question, how? So if I want to explain which capital, let's see, match a state to, well, how about its own capital? And let's see. So there's the first part of the question. Now let's try to explain why that matching is unique. What that uniqueness means is that I'm going to have to make sure that there's only one capital that a state is going to be matched to. Now this isn't a political science class. So if you don't know the answer to this question, well, you should actually know the answer to this question because it's part of the basic knowledge that everybody should have. But the thing that makes this unique is each state has only one capital. So this matching will be to a unique capital. And so that explains both how we're going to do it. We're going to match each state to its capital and why the matching is unique. Each state has only one capital. So the matching is going to be unique. Well, that worked out very well. So let's take a look at this one. Explain how you can match an element of c to a unique element of w. So, well, again, I can start by writing because paper is cheap. And so I want to match those elements of c. So I'm going to match a capital because c set of capitals to a let's see. So w is the set of states. If I have a capital, I need to identify which state I'm going to match it to. How about to the state that it's capital of. So now I've matched my elements of c, the capitals to an element of w. Well, that's some state, but I want to make sure that it's a unique state. I want to explain how I'm making that matching. So when I have the capital, I'm going to match it to the state that it's the capital of. And the matching is unique. Since each capital is only the capital for one state, the matching is unique. So again, I have the uniqueness requires that the state that I'm matching it to be one and only one state. Well, each capital is only the capital for a single state. And so the matching is unique. And now we have to answer this question. Why does this prove cartonality of w is equal to cartonality of c? And because this has that word prove, then it means that we have to look for a definition. So this is, again, another reason why those definitions are so important, because, well, first of all, many questions will be asking about them. But the reality is that the reason that so many questions ask about the definitions is because the definitions are important. I would approve cartonality of w equals cartonality of c. So I need to find a definition that is relevant. So I look around and think about where I've seen cartonality equals cartonality. And so I'll go back and I'll try and find something about that. And here we have it. We have a definition that involves that sentence cartonality of something equals cartonality of something else. And so here's a relevant definition. And it says that two sets have the same cartonality if there is a one to one correspondence between the two sets. All right. What's a one to one correspondence? Well, that means I have to go back and look up another definition. Fortunately, I don't have to go too far. That definition is right before it. A one to one correspondence between two sets occurs when every element of the first can be assigned to a unique element and so on. And let's put these two things together. If I want to prove two things have the same cartonality, I have to show that there is a one to one correspondence between the two sets. If I want to show that there is a one to one correspondence between the two sets, I need to show that there is a matching that matches a element of the first set to a unique element of the second set. Well, I just did that back here. And also a matching that assigns a unique element the other way. And I did that back here. So here is my uniqueness in one direction. Here's my uniqueness in the other direction. When I put these two things together, that gives me a one to one correspondence. And as seen as I have a one to one correspondence, I have equal cartonality. So let's put everything together. And we might say since there is a one to one correspondence between the two sets from A and B, it never hurts to summarize. It never hurts to cross-reference what you've already done. And then the two sets have the same cartonality. And there's our solution to this problem 1.9.