 One of the important ideas in elementary mathematics is known as cardinality, and it appears as follows. Given some set A, then our cardinality is going to be some property that we're going to associate with that set. And so the way we'll indicate that, we'll write cardinality as this N of A, or we might write cardinality by putting the set in between two vertical lines. Now as mathematicians, we care more about how things relate to each other than what they are. And so given two sets, both of which have a cardinality, I might ask the question, well, how can I relate these two? And one possibility, if our two sets may have the same cardinality, if and only if there is a one-to-one correspondence between the elements of the two sets. And so what that means is that I can take every element of A and associate it with a unique element in the set B. And likewise, I can take every element of B and associate it with a unique element in the set A. Now if I can do that, then we say that A is equivalent to B, and we write something like A squiggle B, A is equivalent to B, or I might go back to this cardinality notation. The cardinality of A is equal to the cardinality of B. Informally, you can think about cardinality as a number of elements of a set. However, we're going to be using cardinality G to define what we mean by number. So it's very, very, very, very, very, very, very important to never use the number of elements in a set as part of any formal proof. Now, showing that two things have the same cardinality is actually pretty easy. All we have to do is find a one-to-one correspondence between the two sets. If the set elements are listed, we can simply match every element of one set with a unique element in the other set. So, for example, if I have my set A looking like this, I've listed the elements of that set A, I have my set B looking like this, where I've listed the elements of the set B. What I can do, well, paper is cheap. Actually, pixels are even cheaper, and so I might start down by writing both sets. There they are. And I might, using my creativity and imagination, see if I can match every element of this set with an element of this set. And so let's see how we can do that. I might start with this element A here and say, well, what can I match this to? Well, how about let's make our life easy and just match it to this set element right there? And so we'll indicate that matching using a down arrow to indicate that we're going from this set A to the set B. So I'm going to match this to something down here, and I might match this to that. I might match B to, well, I'm not very creative, so I'll just match it to the run right below it. And likewise for C and D. And so now every element of A has been matched to a unique element of B. I have not matched A to anything else other than this one element here. I could have matched it to this, but I didn't. And so every element of A has gotten a unique match to an element of B. Now I can also reverse the matching. If I went from A down to X, I can also go backwards from X back up into A. So now I've matched every element of B to a unique element of A. And to indicate that I have both matches at the same time, I'll use a double-headed arrow. So there I have a double-headed arrow, and now everything in A is matched to a unique element of B. And everything in B is matched to a unique element of A. There is a one-to-one correspondence between the two sets, and the two sets have the same cardinality. Well, let's take a slightly more difficult example. Let's take the set where S is one of the original. The U.S. states one of those colonies that broke away from Great Britain. And C, well, let's C be the set of things where we're looking at the largest city in each of the original U.S. states. And we want to prove that the cardinality of S and the cardinality of C are equal. So we can try to do the same thing, and we'll rely on our phenomenal memory of history and geography, and I'll put them to the two sets. So my set S, where I'm looking at something that was one of the original U.S. states, well, my memory of history says that Massachusetts was one of them, and New York was another, and Virginia was another, and I don't remember the other colonies. So, but that's okay. Well, let's see what happens. Well, let's see. So now we want to find the largest city in each of the original U.S. states, and I know Philadelphia was a big city someplace. I don't actually, yeah, maybe trying to list these out isn't such a great idea. We have to remember too much about history and geography, so let's see if we can do something a little bit more mathematical and something that requires a little bit less knowledge of history and geography. So let's see if we can describe the relationship. Now, here's a key idea. What we're trying to do is we want to match a element of S to a unique element of C, and conversely, so that we can show that the two cardinalities are the same. So it's helpful to put down some examples to guide our thinking, but here's an important thing to keep in mind. Examples are never proofs. You can never prove anything by giving an example. So we can use those examples to guide our thinking, but at the end of it all, what we have to do is we have to describe our final result in a way that doesn't actually rely on the examples that we've given. So we might try as before, we'll list some of the colonies, and again, maybe I know Massachusetts, New York, Virginia, and some other places. I don't really remember what. And the mathematical way we have of doing, of describing this idea, that there's more here than I'm not going to write, is this ellipsis, this dot, dot, dot. And let's think about this. What I want to do is I want to match a state to a city, and rather than just write down a list of cities as they come to me, what I might try to do is I might try to figure out, well, what would I match the state of Massachusetts to? And after a little bit of thought, we might come to the conclusion that the thing we'd want to match Massachusetts to is Boston. So again, Boston is the largest city in each of the US states, and so Boston is the largest city in Massachusetts. Now, here's the important part of the analysis. Why did we pick Boston? And the logic here is that when you have a state, we're going to match it to the largest city in that state. And so there's a way of generating our one-to-one correspondence. And once we have that rule, once we have this idea that we're going to match each state to the largest city in the state, then, well, if I happen to know what the largest cities are, we can win big on game shows like Jeopardy. So New York, what's the largest city in the time of the colonial era? That's New York City. On the other hand, as long as we know the rule, then even if we don't know the answer immediately, we've described the rule so that when we come to something like this, we can find what we match it to by, for example, looking it up. And so Virginia, it turns out, we're going to match that to Richmond. Now, again, thinking about what we did before, what we were able to do is we were also able to do a reverse matching. Once we matched Massachusetts to Boston, we could also go backwards. And so we can do that reverse matching. And because the correspondence goes both ways, we'll use our double headed arrow. Now, again, it's important because we're trying to prove this statement, we want to connect to the definition. Cardinality of S being equal to the cardinality of C requires a one to one correspondence. So let's describe that every state in S can be matched to a unique element, namely the largest city in the state. Now, here's an important thing. We claim it's unique, but let's go ahead and add a little bit of meat to that and say, there is one and only one largest city in any state. That's what guarantees our uniqueness. Just saying it's unique is important, but I have the I reserve the right to disbelieve your claim without further evidence. Likewise, I want to make sure that I can match every city in C to some state in S. And so every city can be matched to the state that it's located in. And again, we claim uniqueness, but let's go ahead and fill in some of the details. Any given city is only ever going to be located in a single state. And so this tells us there is a one to one correspondence between the elements of S and the elements of C. And so that tells me that by our definition, the cardinality of S and the cardinality of C are the same.