 Alright, so this is going to be a video about nothing, and in many ways this is the most important concept in all of elementary mathematics. And the sentence around something we call the empty set. So the empty set, which we use a special symbol to designate, that's this thing circle with a cross through it, this is a set that contains nothing. What does that mean? Well, the following ideas should be considered equivalent. So first of all I have the empty set, and then I also have, remember I can describe a set by including, by listing its elements in between braces. So here's one brace, here's the closing brace, and I have a set that contains everything that's in between those two braces. Well, there's nothing in between the two braces, and so this is another way of writing the empty set. Well, we've already talked about cardinality, so let's go ahead and say that the cardinality of the empty set, we're going to give this the name of zero, and we're going to give it a symbol, zero. So the cardinality of the empty set is zero, and we write it using our standard zero symbol, equivalently what these two things mean corresponds to the cardinality of the empty set. Now again, our notation, zero is the cardinality of the empty set. However, one thing that I'd be very, very, very, very, very careful with is that zero and the empty set are not the same thing, they are very different things. This is a set, this is a property of a particular set. In fact, all of the following statements are incorrect, even though you may see and perhaps even use them on occasion. If you use them, you are using them incorrectly. Again, zero equal to the empty set, absolutely false, not ever true. Zero, sorry, the empty set equal to the set with the empty set, some sort of notation like this, again absolutely false, and why is that? Well over on the left hand side, this is the set that contains nothing. Over on the right hand side, we have a set. You can tell because we have the two braces, beginning and ending, and what does this set contain? Well it actually contains something, it contains this thing. Now this happens to be the empty set, but it is a thing. And again, even more heinous, zero equals the set containing the empty set, again absolutely false, and this one actually makes no sense, this is kind of like the first statement. What we have over at the left is a cardinality, what we have on the right is a set. Ah, so again here's a nice little problem that we see frequently. In answer to the question, what odd number is divisible by two, you might write something like this. Well it's this, and is that a correct answer? And if not, rewrite it so it's correct. Well let's take that answer apart. The answer here is that this is a set that contains this. So it appears that when we answer this to this question, what I'm saying is that this thing here, this, is an odd number divisible by two. The problem is, this is a set, it's not a number, so it can't possibly be the answer to what number is divisible by two. It can't be the correct answer because it's not even the right type of object. So what's the correct answer to the question? Well I'm going to put in between these two braces every odd number that I can find that's divisible by two. Well I can't find any, so nothing is going to go in between those two braces, and by nothing I mean nothing. There's nothing there, I'm not even going to write something because that is a something in between the two braces. There is nothing in between the two braces. Again, that empty pair of braces can be written as the empty set no braces, so the empty set is another way to indicate the answer. Now here's the surprising thing. The empty set allows us to define all of the whole numbers. So remember we have the following definitions, the cardinality of A is equal to the cardinality of B. If and only if A is equivalent to B, likewise the cardinality of A is less than or equal to B, if and only if A is equivalent to C, where C is a subset of B. And we've already defined that zero is the cardinality of the empty set. So now we're going to define all of our whole numbers. So we're going to start as follows. We're going to let A be the cardinality of some set, and I'm going to take some other set B. And if every proper subset of B has a cardinality that's less than or equal to A, and again I've already defined what that means, and there is a subset of B with cardinality equal to A, then I'm going to define the following. A star is the cardinality of B. Now what is this A star? Well I interpret this. This is the next higher cardinality. This is the cardinality after A. This is the one that follows A. And now we're going to prove something very exciting. We're going to prove that this set here has cardinality two. And again in plain terms we might say without mathematics there are two things in this set, and the question is well how do you know that there are two? Well you can't really just count them one two, because unless you know what two is that doesn't really help you. But we are now in a position to be able to prove that this set has cardinality two. Now to prove this let's go ahead and invoke our definition of cardinality, and again the idea here is that if every proper subset has a cardinality less than or equal to something, and there is a subset with cardinality equal to that something, then our cardinality is the next higher cardinality. And in this particular case the set that we're looking at is this set xy, I'll replace that, and what I want to show is that this set xy has cardinality two, and that means that this is two is the next higher cardinality, so that tells me that my original cardinality has to be equal to one. So there's my setup for what the proof is. Now again the important thing to notice here is all I've done is I've taken the definition of what cardinality is, and I've just replaced it with the particular set that we're working with with what we're trying to prove, and one other detail here which is that there's one other item which is the preceding cardinality that I had to fill in. Now in terms of finding a proof, here's a good guideline to go with. Anything that is introduced as part of the definition is something you don't have to say too much about. In particular something here that is part of the definition, which is that if every proper subset of xy has cardinality less than or equal to one, what that means is if I look at a proper subset of xy of this set, I can identify its cardinality, I can identify that the cardinality is less than or equal to one, and because it's part of the definition I'm invoking, I'm not going to have to spend a lot of time explaining why that is the case. Now if it weren't part of the definition, I couldn't just dismiss the problem that easily, but because it is part of the definition I don't have to spend a lot of time on it. So let's take a look at those proper subsets. Again the concept of proper subset is part of the definition, so I'll go ahead and just say here's a proper subset, its cardinality is one, again the cardinality one is part of the definition, I don't have to say anything about why I know this cardinality is one, I don't have to say anything about why I know that this is a proper subset. Well here's another proper subset, again cardinality one, and then the subset that contains nothing, the empty set is always the subset of any set, and by definition the cardinality empty set is zero, and so do I have. Well these are all of the proper subsets of this, and they all have cardinality less than or equal to one, which means that this set I've been working from has the next higher cardinality and the next higher cardinality we call two, and there's our proof that the cardinality of the set is two, and again quick observation here, the answer to the question prove this statement is everything we have in green.