 So fine diagrams are a nice visual aid towards the understanding of sets. And so it's a visual representation of a set where we represent a set by some sort of closed plane figure, a circle, a square, anything that has an inside and an outside to it. And anything that's inside the figure belongs to the set. So this green dot is an element of the set. Meanwhile, anything that's outside does not belong to the set. So this red dot out here does not belong to the set. Now, when you draw a Venn diagram, you should view your drawing as a challenge, as an open challenge to anybody who comes along. And in particular, you should view this as following. So when you do a Venn diagram, you should assume that your worst enemy is going to come along and try to show the world that your Venn diagram doesn't work. And what you want to make sure of is that your worst enemy can't do that. So for example, let's say we want to draw a Venn diagram showing two sets A and B, where every element of A is an element of B. And so I'll draw my set A, and then I'll draw a second set B. Well, maybe I'll draw something like that. Alright, now I leave this out, and then my worst enemy comes along and tries to show that this Venn diagram is wrong. And it's pretty easy to do that. They can say, ah, what about this point here? This is something that is in A, but is not in B. And you're supposed to draw a Venn diagram where every element of A is an element of B. Well, here's something that is not an element of B. So your Venn diagram is wrong. And because of this, you abandon mathematics and go join the circus. Well, actually we could redraw the picture. Let's not do something quite so drastic. Let's redraw a picture. So there's our set A again. And well, maybe I'll draw another possibility for B. And maybe this works. Well, okay, so my worst enemy comes along and tries to show the world that the Venn diagram that we've just drawn is wrong. And they say, well, what about this point? Well, nope, that doesn't work. This is something that's in A. And it is an element of B because it's inside the B circle. So that doesn't work. And here's something. Nope, that doesn't work either because this is not an element of A. And I don't know anything about that. The only thing I know is that every element of A has to be an element of B. If it's not an element of A, I don't know anything about that. So doesn't matter. And likewise, if I'm out here, this is not an element of A. So what the relationship of this is to anything else is completely irrelevant to this particular problem. So they can't show that there is anything that an element of A is not an element of B. And so they have failed to show that this Venn diagram doesn't work. And so your worst enemy goes to say, you know, I guess this is all right. And there's our Venn diagram.