 So another thing you can do once you have two sets is to define a one-to-one correspondence. A one-to-one correspondence exists between two sets if there is a way to match every element of either set to a unique element of the other set. For example, if possible, find a one-to-one correspondence between these two sets. Definitions are the whole of mathematics. All else is commentary. We want to find a way, if possible, to match every element of either set to a unique element of the other set. And so we can try to match every element of A with a unique element of B. So we might list the elements of A. Well, let's just match up the elements and see what happens. We can match one to R, two to B, and three to G. And so we've matched every element of A with a unique element of B. But we also have to match every element of B with a unique element of A. Fortunately, we can do that pretty easily. We can do that using the same matching just going backwards. And so here is A one-to-one correspondence between the two sets. So a useful strategy. Try to do the same thing a different way. So let's try to find another one-to-one correspondence. Again, we'll take the elements of the set A and match them up to the elements of the set B, but not in the order R, B, G, but maybe in the order, oh, how about G, R, B. That matches A to B, and we can match B to A by going backwards. How about these sets? So we can try to match every element of C with a unique element of D. So let's list our elements of C. And we'll match, well, how about spade to empty set and club to one. But if we try to match every element of D with a unique element of C, well, we can go backwards, empty set to spade and one to club. But if we try to match every element of D with a unique element of C, we can't, because two doesn't have anything it can match to that isn't spade or club. Or can we? And this may be the hardest thing in mathematics. Remember, be your own harshest critic, because if you're not, someone else will be. We tried one time to find a one-to-one correspondence and failed. But if you quit after your first failure, you will never succeed. Maybe something about our initial correspondence prevented us from making a one-to-one correspondence. So what if we started with the set D? Well, let's try to match every element of D with a unique element of C. So we'll list our elements of D, and we'll start matching them up with elements of C. So how about empty set to spade, one to club, and two to... And so this seems to be impossible. Actually, we can't conclude it's impossible until we check every possibility, or we develop some more mathematics. And so let's consider our sets again A, B, C, and D. A and D do have some elements in common, but other than that, the sets have no overlap. But in fact, three of these sets do have a property in common, and we might call it property threeness. Which is to say, these sets have three elements, and this one does not. Definitions are the whole of mathematics, and we might define... A set has three elements when it has three elements. Yeah, that doesn't work, because we're trying to define something in terms of itself. It's a circular definition. It's like saying blue is the color of things that are blue. Well, if we can't define three, maybe we can at least give a name to the property that we're talking about, so maybe we'll define something called cardinality. The cardinality of a set designated this way is the number of elements in the set. However, while we might do this, later on we'll try to define number, and we'll be using cardinality in that definition. And what that means is we can't use this definition. Again, that would give us a circular definition by trying to define something in terms of itself. It's better to think of cardinality as a property of a set. And again, what matters is not what cardinality is. What matters is what cardinality does. So what does it do? And we define two sets have the same cardinality if there is a one-to-one correspondence between the elements of the two sets. If that happens, we write cardinality of a is equal to the cardinality of b. For example, we might have these two sets, and we'll show that the cardinality of a is equal to the cardinality of b. And remember, we already found a one-to-one correspondence. Definitions are the whole of mathematics. All else is commentary. If there is a one-to-one correspondence between the two sets, then the two sets have the same cardinality. And so we can say that the cardinality of a is equal to the cardinality of b.