 So let's talk a little bit more about cardinality. To that end, it helps to introduce the idea of a proper subset. So remember our definition, A is a subset of B if every element of A is also an element of B. Now, this can actually lead to some peculiar situations because mathematicians are absolutely, literally minded. And in particular, if you say that A is a subset of B, if this is true, then regardless of what it looks like as long as this is true, we have a subset. So let's take the set B consisting of these things and I can form a subset by taking some of these and copying them into another set. So let's take a look at that. So I'll take my set A and I'll copy some of the elements. And so if my set A consists just of that, then A is a subset because everything in here is in our set B. If I throw in something else then again, everything in A is in set B and so A is still a subset. And if I throw in something else, everything in here is in here. So A is a subset of B and so this is still a subset even though they are the same set. Now, this doesn't bother mathematicians because again, mathematicians are literal minded about things but we may want to include a few more nuances. So we'll introduce the idea of a proper subset. So something is a proper subset of another if, well, first of all, it has to be a subset but what makes it proper is it's not the same. This is a subset of B but it's the same as B so it's not a proper subset. On the other hand, if I were to exclude one of these elements then I would have a proper subset. Now, once we have this notion of proper subset we can define the whole numbers in the following way. The whole numbers are the cardinalities of sets and we're going to define them as follows. So the number zero, we're going to define this as the cardinality of the empty set. The cardinality of one, the number one is the cardinality of any set whose proper subsets all have cardinality zero. If I want to take a look at the number two I'll define that as the cardinality of any set whose proper subsets have cardinality zero. Well, I've defined it back here or one which I've defined back here. For three the cardinality is going to be any set whose proper subsets have cardinality zero, one or two. And if I'm going to define the number four well it'll be the cardinality of any set whose proper subsets have cardinality zero, one, two, or three. And in general if I want to define any whole number it's going to be the cardinality of any set whose proper subsets have cardinalities equal to any of the preceding numbers. So for example let's take this set B which consists of this thing and let's actually prove that the cardinality of B is equal to one. And so here's a wrong answer. B has one element so the cardinality of B is equal to one. And this is wrong. This is not an answer because it has no connection to the definition of cardinality one. So it has no connection to the definition it is not a proof. If I want to prove it I have to go back to my definition of cardinality. What does it mean that something has cardinality one? So here's my definition of the whole numbers again. The number one is the cardinality of any set whose proper subsets have cardinality zero. So I'll copy that portion of the definition down for reference. Cardinality is one if the proper subsets have cardinality zero. So let's take a look at what those proper subsets of B are going to be. And we'll form those by copying some but importantly not all because then they wouldn't be proper of the elements of B into a new set. So I'm going to form a set A and I'm going to take an element of B and I'm going to copy it into... Well the problem is if I copy this element into A then A and B are going to be the same set and A will be a subset but it won't be a proper subset. So this is the only proper subset of B. This is the only thing that I could form where everything in here is an element of B. Well there's nothing in here so automatically everything is true about nothing. So here's my only proper subset of B. It's the empty set and by definition the cardinality of the empty set is zero. And so let's connect this to our definition. The proper subsets of B have cardinality zero. So B has cardinality one. So again our definition cardinality is one if the proper subsets have cardinality zero. Well here we have it. Proper subsets do have cardinality zero so our set has cardinality one. Well now we're in a position to compare cardinalities except there's a little bit of a problem and I'll ask it this way. Have you stopped playing Angry Birds when you should be doing your homework? Well as a general rule what happens is it's easy to prove a positive. You can prove you've done your homework by showing someone your homework. On the other hand it's very hard to prove a negative. Can you prove that you weren't playing online games when watching this video? And here's where the difficulty arises. If we can find a one-to-one correspondence between two sets we know that they have the same cardinality but if we can't find a one-to-one correspondence maybe we weren't clever enough to find it. So this suggests the following. Say I have two sets A and B. If I can find a one-to-one correspondence between A and a subset of B and that's easy to find if it exists then the cardinality of A is less than or equal to B. On the other hand if I can go further if the cardinality of the one-to-one correspondence exists between A and a proper subset of B then the cardinality of A is strictly less than the cardinality of B. And that means if I can find a one-to-one correspondence between some sets I can start to answer questions about cardinalities. So for example let's take these two sets A is this thing, B is this thing and I want to prove that the cardinality of A is strictly less than the cardinality of B. Well here's a wrong answer. A has three elements, B has four elements, three is less than four. Therefore well this is not a proof and again this is not a proof because we have to rely on deductions from the definition. There is nothing in here that refers back to the definition of unequal cardinalities. So what we want to do is we want to show there is a one-to-one correspondence between A and a proper subset of B. Well since we have the elements listed we might try to find a one-to-one correspondence. So here's our elements of A, here's our set B and let's see. So I can try to match them up so maybe I'll match A to x hat, I'll match B to y tilde, I'll match C to z check and well I don't know what I'm going to do with this w dot but I don't actually care because the thing that I've noticed is I can connect to my definition. I have shown there is a one-to-one correspondence between the elements of A and a proper subset of B. So this collection here, this is a subset of B but it's not the whole of B. So here I have my one-to-one correspondence between A and a proper subset of B and I can identify what it is and so that tells me by my definition that the cardinality of A is strictly less than the cardinality of B.