 Now let's do some example exercises. We're going to stick to proofs for now. We've got to prove the associative property of the intersection of three sets. So if I have the intersection of A and B, it's intersection with C. I'm saying that would be the same as writing A's intersection with the intersection of B and C. How do we do that? Well, we take out what we have in our little box. The definitions that we've had before, and these are the two I want you to concentrate on. This is in our box. As human beings, we've decided that the intersection of two sets would be the set of all elements X such that X is an element of A sub 1, the first one, and X is an element of A sub 2. That's our definition. We've decided that. There's no need to prove that that is our definition. That is how we define it. And we said that two sets are equal if and only if one is a subset of the other and the other is a subset of the first one. That's in our box. Now, let's prove this. The way that we're going to go about it is just to start somewhere. We let X be an element of this first part of the equality. A's intersection with B's intersection with C. If that is so, let X be an element of that. Then X is an element of A's intersection with B and X is an element of C. All I've done now, I've used my definition. I've only used my definition. Then that is so, so that X is now also an element of A and X is an element of B. There, I've just unpacked it again and X is an element of C. Now, if X is an element of A and X is an element of B and X is an element of C, let's use some of the scenarios. I'm going to say if X is an element of B and X is an element of C, then what follows? Follows by the definition that X is an element of B's intersection with C. The other scenario now is now this. Then X is an element of A and X is an element of this. B's intersection with C, so that X is now an element of A's intersection with B's intersection with C. Now, it might not look that satisfying because what I'm doing, I'm actually just translating what I have there into language, into words. But be careful. What I'm doing is I have taken an arbitrary element inside of this first one and I'm unpacking that by definition. That is all that I'm doing. So if X is an element of that and X is an element of this now, that means this must be a subset of that. So A's intersection with B's intersection with C must be a subset of A's intersection with B's intersection with C. This must be a subset of that because I've started off with it being there. I've let it run through which now says that X is an element of this. X started here, so it means X must now be a subset of that. Now we can write conversely because what we have here is we've got to prove it in one direction and the other direction. We must also prove that this is equal to that. And how am I going to do that? Well, I'm going to say let X be an element of the second part, A's intersection with B's intersection with C. If that is so, then by definition, by our definition X is an element of A and X is an element of B's intersection with C. So that X is now an element of A and this means X is an element of B and X is an element of C. Now we have this scenario that X is an element of that and that and that. Let X be an element of these first two. If X is an element of A and X is an element of B, then by our definition it holds that X must be an element of A's intersection with B. That means so that X is now an element of A's intersection with B and X is an element of C. X is an element of C. And if that is so, by our definition that means that X is an element of A's intersection with B, it's intersection with C. So I see we started off there and now we are here. That means therefore we have that A's intersection with B's intersection with C must be a subset of this because X was an element of that first one. It is now an element of this so this must be a subset of that. Of A's intersection with B's intersection with C. Have I got that right? So this must be a subset of what we have here. And now we are going to use this second part. So if the one is a subset of the other and the other is a subset of that, therefore by this definition, therefore we have proven that A's intersection with B's intersection with C equal A's intersection with the intersection of B and C. There we go. So we have just used some of the definitions that we have had that we can take out. This can be proven. This is our definition. We are taking that out. It lives inside of our box. We can use them to do our proof. All we must remember is that what we are suggesting here is this implies the other. A's intersection with B's intersection with C implies that A's intersection with the intersection of B and C. So I have got to prove it in this direction and I have got to prove it in the other direction because what I have proven here is that this is a subset of that. What I have proven here is that this is a subset of that. And because they are subsets of each other, I can use the fact that they are now equal to each other.