 So our basic operation of arithmetic is addition, and in general, let's talk about what mathematics is. Mathematics occurs when we make abstraction of some sort of process. So in this particular case, sometimes we put things together. So maybe I'm going to take this thing here, and I'm going to take this thing here, and I'm going to put them together. So I put them together, and it's kind of obvious that if I put this and this together, I make this thing. And that's a really obvious thing. That is a process. Now when I do mathematics, what I'm going to do is I'm going to identify some abstract feature of this process, and that's going to be our mathematics. And we can choose what we want, and in this case, maybe the feature we'll choose is we'll take a look at the cardinality. So I have a collection of things here. I have a collection of things here, and when I put them together, I get a collection of things. And maybe the thing I'll focus on here is the cardinality of these collections. So this first collection, well, has this property of cardinality 4. The second thing has this property that it has cardinality 3, and when I put these things together, what I have is a thing, and I looked at cardinality, I looked at cardinality, I'll continue to look at cardinality. This thing has cardinality 7. And this leads to what we can think about as an addition for plus 3 is equal to 7. Now here's an important thing. This is, and should be, a very, very familiar fact. But again, make sure you understand the difference between what is familiar and what is easy. This is not easy. This is an extremely difficult concept. This is extremely hard to understand. It makes no sense unless you've learned quite a bit. On the other hand, it is really easy to see that this and this make this. Here's something that's easy. Here's something that's familiar. Well, let's try and put a little bit of more formality on this. So what did we actually do? Well, what we did is we took two sets in this particular case. We took two sets, and the important feature that we're going to add is the intersection has to be the empty set. So I'll take two sets whose intersection is the empty set, and each set has this property of cardinality. A has some cardinality, B has some cardinality. Then when I define addition A plus B, well, that's again a cardinality of what we get when we put the two sets together. Now here's a useful idea. If I'm going to use this definition, say in a problem that requires us to prove something, if you're going to use a definition, anything that's part of the definition, you can simply state without having to get any further comment. You don't need to defend what you are saying. You can just simply state it, claim it, and hope, and without any further comment, assume that it's true. So for example, cardinality of A, cardinality of B, this is part of the definition. So you can simply say, I have this set A, and its cardinality is whatever I need it to be. I have some set B, its cardinality is whatever I need it to be, and you don't have to say how you know what the cardinality is. Likewise, the two sets having an empty intersection, you can simply say, the intersection of the two sets is the empty set. Again, the cardinality of the union is part of the definition, so you can simply say, I know what the cardinality of the union is, it's this value. Now, if you need those things, you can simply claim them without having to give any elaboration, and do make sure that you're actually claiming something that is true. Don't write down a set and give the wrong cardinality, don't write down two sets that are non-empty, because that invalidates your work. So for example, let's say I want to prove that 2 plus 3 equals 5. Again, point of emphasis, the only way you can do this is to use the definition. There is no other way to answer a question that says, prove, you must go back to the definition. So, let's put that up for reference, let A and B be sets, where A intersects B as the empty set, A is the cardinality of the one, B is the cardinality of the other, and A plus B is the cardinality of the union. So, that's what I need, I need 2 plus 3 equals 5, I need a 2, I need a 3. So, what that says is I need two sets, so the cardinality of A is going to be 2, that's my first term, so I need a set with cardinality 2. So, I'll write down a set and it has cardinality, whoops, that does not have cardinality 2, so let's fix that. So, there's my set, it has these things in it, the cardinality of that set is 2, there's part of my definition, right there. I need another set where B, my second number, 3 is the cardinality of that set. So, here's B, it's this set, and 3 is the cardinality of B. And again, because this is part of my definition, I don't have to say anything about this other than make sure that it's actually true. Let's see, I need A intersect B to be the empty set, oh wait a minute, these two do have something in common. I can claim A intersect B as the empty set, but it's not true, so that would make the rest of the work meaningless. So, let's fix it, little change B to something else, so now B has things that are entirely different from what A is. So, now let's go ahead and put those together, let's fill in the rest of our definition. I have two sets, here, here, A intersect B as the empty set, check, A is the cardinality of the first set, 2 is the cardinality of A, 3 is the cardinality of B, so 2 plus 3 by definition is going to be the cardinality of the union. So, let's go ahead and look at that union, let's see, so that's the cardinality of the union of the two sets, that union is going to be formed by taking everything in A together with everything that's in B. So, here's my stuff from A, I'm going to take stuff from E, drop it down into there, and again, because the cardinality is part of the definition, I can simply say the cardinality of this set is what it should be, 5. And there's our proof. Now, here's an important question, what do you need to write down to answer this problem of proof 2 plus 3 equals 5? The thing you need to write down is everything here, all of this is part of the proof. If you miss any portion of this, if you skip any of it, if you leave anything out, you do not have a complete proof. In order to answer the question completely, everything here has to be included. So, this is...