 All right, so let's take a look at problem 1.6. And here's an important idea. If you want to learn anything about anything, you'll need to read about it. Watching videos, attending lectures, that sort of thing, those are very helpful. But the printed word conveys far more information in a far smaller space. Consider how long it took me to convey the information in this one paragraph, versus how quickly you could read the paragraph. Now, multiply that by 1,000, and you'll have an understanding of why the printed word is so very important. So here's what the printed word has to say about the Venn Diagram. So this is from your book. And this describes the Venn Diagram. And one important thing here is this sentence at the very end. You should always assume that there are other elements in and out of the set that aren't shown. And what that means is if I draw a set this way, I have to assume there are elements over here in the outside and there's elements all over the set inside as well, which means that the elements I see, and even if I draw in some elements, the elements I draw in aren't all that there are. And there are going to be, we should always assume, there are other elements that we're not drawing in every place that they can be. And this has some important implications. Now, if you really want to understand Venn Diagrams, don't watch the rest of the video. You can't learn mathematics by watching somebody else solve problems. So again, if you want to learn about Venn Diagrams, solve this problem. If you don't want to learn about Venn Diagrams, you could watch somebody else solve the problem. So let's take a look at this. So draw a Venn Diagram with some sets where A and B can never have column and elements. And every element of C is also an element of A. So to begin with, we'll draw two sets A and B. So we'll draw a circle representing a set and see if they meet the requirements. So I'll draw two random circles like this and let's see. So I have to assume there are elements someplace, every place, even if I don't show them. So if I draw the circles this way, the thing to notice is that if I put an element of A, if I put down an element in A, that is automatically going to be an element of B. Except A and B are supposed to never have common elements. So I know that this diagram can't possibly be correct. So we should try a different drawing. So I'll draw a different set of circles. So I'll draw something like this. And while here I can put something in here that is not an element of B, but because I should assume there are elements even if I haven't drawn them, I could put an element right here. And that element, if I put an element in here, that is in A, but it's also in B. And it's a common element of A and B. And I know that A and B could never have common elements. So I'll try a different circle. Well, what if our circles look like this? So if I put something in A, no matter where I put it, it can't be in B. And if I put something in B, no matter where I put that element, it can't be in A. Since it's not possible for something in A to be in B, or vice versa, then I have drawn two circles of N diagram that shows that A and B can never have common elements. Well, let's throw in that third requirement. L for element of C is also an element of A. So let's draw a third set C. And to include both, I'll draw this ellipse. And so maybe it looks something like this. I'll draw something like that and check to see if it meets the requirements. Every element of C is also an element of A. Well, the problem is, again, I can put something any place in here. I have to assume that there are elements even though I haven't drawn them. If I put an element in C, that is not going to be necessarily an element of A. So this drawing doesn't work. Well, I'll try a different drawing. So maybe I'll draw something like that. And if I put something in C, well, if I happen to put it here, then it's going to be an element of A. But I could put it here because, again, even though you don't see them, you should assume that there are other elements. So this element, if there's an element here, it is not an element of A. So this doesn't fit that requirement. Well, maybe something like that. And if I put something in C, any element of C can't be an element of A. So this doesn't work. And so that means we should try a few other possibilities. Maybe it looks like this. Well, now let's see if I do that. Well, it fails. Every element of C is an element of A. In fact, no elements of C are going to be an element of A. How about over here? Again, I have to assume that there are elements even if I don't draw them. So maybe there's one over here, which is not an element of A. And so again, it fails this requirement. Every element of C is also an element of A. And maybe I draw something like that. And that seems to work. If I put an element in C, that's automatically going to be an element of A. And let's go ahead and check. A and B can never have common elements. Figure that out. Every element of C is also an element of A. That also works. And it looks like this is a good diagram.