 I'm Zor. Welcome to Unisor Education. I'd like to offer a couple of exercises on the SET theory. You probably have to listen to the lecture about the SETs, which is preceding this particular problem, and I would like to present something which is called a problem number one in the SETs here. It's a very easy problem. It's not really a problem. It's really an exercise. It will probably help you to understand what exactly SETs really are. Let's consider a set which consists of the following elements. Let's say football, planet, book. Question number one, how many elements does this set have? Well, obviously you can answer that question yourself. You don't do any exercise. So first is how many elements? Well, number two, consider the subsets of this particular set. If you can actually point to this is the subset and this is the subset, that would be great. So question is what subsets? Question number three, if I will ask you which elements are man-made, you should be able to basically tell me which are man-made, which will be a subset, obviously. Well, that's about it for this very, very simple exercise. So you can pause right now, you can answer these questions yourself, but obviously I'm going to answer them now. Well, obviously there are four elements. This is not really a question. Second question is a little bit more interesting. When we are talking about subsets, we really have to enumerate all of them. Well, let's try. We have four different elements and it's not really easy to enumerate all the different subsets which this particular set contains. But let's try. Obviously every element by itself constitutes a subset. So first of all we have four subsets, each of them constituting one and only one element. All right? So let's put it here. First is football, second is planet. And you know what? I will put it in curly brackets. So you will distinguish them from the elements themselves. So these are subsets which contain one and only one element, a book and a person. Obviously we can combine these elements into pairs and each pair would be a subset by itself in its own right. Well, there are many different pairs here. Actually there are the following one. Five. Let's combine football with something else. So it would be a subset which contains football and planet. Six. Football is seven. Football and person. These are all the pairs which contain football. Now which pairs contain planet? Well, planet and football we have already counted so we have to go only down. So it will be planet and book. Counted everything which contains planet, everything new actually which contains planet. Now everything new which contains book is only the book with a person. Well, as you see it's not really easy to enumerate all of them. We really have to go through all the combinations. And with the person we have already counted these are all old ones. So these are single element subsets. These are two element subsets. Well, are there any three element subsets? Obviously. Now the easiest way in this particular case to count all the three element subsets is to count which element does not belong to that particular subset. So instead of counting these three elements I would say that there is a subset which does not contain person. Instead of counting let's say this element I would say this is the subset which does not contain football. There are only four elements, there are only four such three elements subsets because we can exclude only this or this or this or this. So I will write it down the following way. This is a subset which does not, does not, a negative, a negation, does not include football. This is a subset which does not include planet. Then the book, these are three element subsets which this particular set contains. Okay, one, two, three, and there is only one remaining subset which contains all of them, which is the whole set. So 15 is basically the whole set of four elements. Have we counted all the subsets? Well, the answer is no. There is one more subset which we have not counted. This subset always, is always present in any kind of set. This is a subset which I'll start with number zero which contains basically empty set or a subset. So empty means does not contain anything. So this is a subset which contains zero elements from the original set. These are all the subsets which contain one. These are which contain three. These are which contain, sorry, two. These are three and this is the only one which contains all four. So we have numbers from zero to 15. So we have 16 subsets and that's the answer to question number two. This set contains 16 subsets. And which are man-made? Well, this is very easy. Obviously, football and book are man-made. So there is only one subset which contains man-made elements which is the subset which contains football and book which is number six in this particular enumeration. Okay, this is it for a very simple exercise. And try to continue. I will put an exercise number two, number three. I don't know how many it will come up with. But anyway, this is it for right now. Thank you very much.