 Let's have a chat about the ordering of sets. Now, many times you'll just see we use a set 1, 2, 3, 4, 5, a, b, c, d, and you think, well, you attach some extra meaning to it, because it's in normal terms, we do see an order in the natural numbers and we do see an order in the alphabet. But that is not what is implied. You have to explicitly state that ordering. Don't just assume it because in human terms we naturally give an order to that. So I'm going to use some binary relation. So we all know what relations are now. And the first relation that I'm going to look at is the relation. And I'm just going to use our full relation again is the following is less than or equal to, less than or equal to. So that is my relation. And I just look at my set on the set of natural numbers, on the set of natural numbers. Now, that relation is going to force some order on that. Because if I look at my set of natural numbers, that being 1, 2, 3, 4, 5, etc. It's going to bring some order because 1 is less than or equal to 2. 2 is equal to less than 3 and 4 and 5, etc. So it's going to bring this natural order to it. But if I look on this R being R is this divides and we can write that. And let's have my set on the set. I'll show you now. I'll be a bit confused. I'll show you now. On the set, we're going to have my set A. And my set A is going to be, let's make this set 1, 2, 1, 2, 3, 4 and 12. Does that bring order to the set? Not totally because 1 does divide 2. Remember, divides means I divide 1 into 2 or 2 divided by 1. That's 2 and there's no remainder. So 1 divides 2, 1 divides 3, 1 divides 4 and 1 divides 12. But 2 does not divide 3. 2 does divide 4 and 2 does divide 12. 3 does not divide 4, but it does divide 12 and 4 divides itself and 12. So you see there isn't this order to that. So there is a way that we can write this ordering. And by the way, so this is called partial ordering. And the way that we can do this is that we have this 1, 1 divides 2, 2 divides 4, 4 divides 12, 1 divides 3 and 3 divides 12. So there is some partial ordering. Whereas if my relation on this was less than or equal to, my ordering would have been 1, 2, 3, 4, 12. And this is a bit ugly. But anyway, that would be total ordering of this relation on this set. On this set, if my relation is less than or equal to, it would be this. And this would be partial ordering though, because there is this out of step here. So let's have a look at that. Let's have a look at some examples. Or let me let's add some new terms here. So this first one here, let's have all these terms. Let's see where I put them down. Let's see where I put them down. We have a first element. We have a first element here. And let's have a look. I think I've lost it somewhere in all of my notes. Let's see. There we go. So we have a first element. We have a last element. So here we will also have a first element and we will have a last element. So that's easy enough to see on this relation on the same set A and on this relation same set A. So for this one, we would say we would define this first element. If we have the following, that we have A, R, X for all, for all of these X in element of A. Now think of that. So this one is for the division of the set. So that the first element 1, 1 divides 1, 1 divides 2, 1 divides 3, 1 divides 4 and 1 divides 12. It divides all of them. So that would be the first. So this one, the last one, it will have the following. X, R, G. And that means that all the elements here, we can write as divides 12. 1 divides 12, 3 divides 12, 2 divides 12, 4 divides 12. They all divide 12 and therefore it becomes the last element. But there's also something called the minimal and the maximal element. So this is first element and last element. Element. And then we're going to have the minimal element and the maximal element. Minimal element and the maximal element. By the way, there might be something that you are asked to prove that if these exist, if these exist, they are unique first and last element. But let's have a look at the minimal and the maximal element. And that's where, yes, here is my two sets that I want to show you. So if I have the following, I have my partial ordering being A, B and there I have C and D. And I have the following one A and B and they both go to C and C goes to D and then C goes to E. But D also goes to E. Then you can clearly see that there is this more than one maximal element here. And there's a single minimal element. And here we have a single maximal element and two minimal elements. So I think it's quite natural when you look at something, when you do look at something like that, what minimal and maximal elements are. Just a few terms then for you to remember with the ordering of sets, we can have this partial ordering and we can have this complete ordering. But there is this first concept of a first and the last element and there are these concepts of minimal and maximal elements when we talk about ordering.