 Mathematics begins with the process of counting, so let's think about what this process actually is. And so mathematically, formally, what we're looking at is that the number of objects in a set is a particular property of the set, which we call the cardinality of the set. And it's some property of that set, and again in modern mathematics, we don't necessarily try to define what something is, but rather how it relates to other things. So again, you can think about this cardinality as being the number of things in the set, but again we don't really want to use that term. We'll indicate the cardinality of the set. There's two ways of doing this. We either indicate it N of A, which kind of suggests the idea that this is the number of things in A. As we'll see, notation persists because it reminds us of something useful. The other way we indicate cardinality is to throw our set name inside two vertical bars. Now again, we don't really care what things are. We do care about how they relate to each other, and so here's the key idea of cardinality. Two sets A and B have the same cardinality, if and only if. This is a definition. It works both ways. There is a one-to-one correspondence between the elements of the sets. What does that mean? Well, what it means is that every element of A can be associated with a unique element of B, and every element of B can be associated with a unique element of A. Now, because mathematicians like to talk, we'll throw in a couple of other equivalences, we'll say that if we do have two sets with a one-to-one correspondence, well, they have the same cardinality. We also say that the set A is equivalent to the set B, and we're going to write either this form here, A squiggle B, A is equivalent to B, or because we have just defined them as having the same cardinality, we're going to indicate that the cardinality of A is the same as the cardinality of B, and then again, informally, we can think about both sets A and B as having the same number. Now, here's a very important thing to notice. At no point have we ever counted the number of elements in the sets A and B. Well, how do we then count things? Well, let's take a look at an example. So, here's a good example that we can take. Let S be the set of things that are U.S. states, and let C be the set of things that are the capital cities of U.S. states, and we want to show that the cardinality of S is the same as the cardinality of C. Now, before we go on, let's consider what we might think about as a bad solution. And a bad solution looks like this. There are 50 U.S. states, there are 50 U.S. capital cities, so the cardinality of S is the same as the cardinality of C. And the reason that this is a bad solution is that this has nothing to do with our definition of cardinality. We're saying that the, we want to show that the cardinality of S is the same as the cardinality of C. If you're going to show something in mathematics, what you want to be able to do is refer back to what it means for these two things to be the same. So, let's get rid of that bad solution and put in a good solution. So, let's think about this. We want to show that the cardinality of S is the same as the cardinality of C. And so, it is often the case that it's useful any time we do a show problem in mathematics. Let's consider what the definition is. So, going back to our definition, the equality of our two cardinalities, cardinality of S, is the same as the cardinality of C. What we want to do is we want to show that every element of S can be associated with a unique element of C and also that every element of C can be associated with a unique element of S. And what this does is this establishes the one-to-one correspondence between the sets S and C. And as soon as that one-to-one correspondence exists, we know that the two sets have the same cardinality. So, thinking, thinking, thinking. Well, let's start off with this side. So, every element of S, well, S is the set of U.S. states. So, I'm going to pick an element of S and pick an element of C, which are the capital cities of U.S. states. And what I'd like to do is I'd like to associate a U.S. state with a specific capital city of a U.S. state. So, thinking, thinking, thinking. Well, well, it's often easier to organize our thoughts if we take an example. Let's take the state of, oh, I don't know, New York. If I want to associate a state such as New York with a unique capital city, well, what would I do? Well, I would probably associate the state of New York with, let's see, what's it, the Tallahassee. That doesn't really make sense. Ask any almost normal person to do this, which is to say, you know, almost everybody. And if you're going to match a state to a capital city, it would be reasonable to match the state to the capital of that state. And you may want to think about this for yourself. Is it true that that state capital is unique? Well, sure, every state has just one state capital. Now, we have to also go to the other side. Every capital city has to be associated with a unique element of S, a unique US state. And so again, we want to match a capital city of a state to a state. So if I were to take the city of, oh, I don't know, Tallahassee, and which is a capital city, this is not a geography class. So if you didn't know, Tallahassee was a capital city, I'll tell you that it's a capital city. And let's see, if I want to associate that with a unique state, how about Oregon? Well, much more likely a normal person would say, well, we might as well match the capital of the state to the state that it's the capital of. And again, this is not a geography course, but points if you remember that the Tallahassee is the capital of the state of Florida. So there I'm going to match a capital city to a unique US state, namely the state that it's the capital of. And every US state gets matched to a unique capital city, namely the capital itself. And to tie it all together, we throw down a concluding statement and we say something like, well, there is a one to one correspondence. Here it is. And the important thing is that all of this is part of the solution. Since there is a one to one correspondence, then the two sets are equivalent. And the cardinality of the first set is the same, is equal to the cardinality of the other one. Now, one important thing to note, again, to emphasize, we did not come to this conclusion by talking about the number of states and US capitals. What we did is we showed that the two sets were related in this way that gives us the one to one correspondence. And the significance of that is if I change the question very slightly here, let S be the number of states in 1850, and C be the set of capital cities of US state in 1850, you'd still be able to answer this question even if you didn't know how many states there were in 1850. Which again, emphasizes a very important idea here, that showing that two sets have the same cardinality is independent of knowing how many things there are in either set.