 I'm Zor. Welcome to Unisor Education. Right now, we will go through an exercise number two for set theory. So, problem two, and this is about sets. Let's consider two different sets. Set A, which contains wall, building, planet, and set B, which contains, let's say, wall. This small exercise will be about certain operations which you can do with these sets. So, I will do the following thing. I will do the union. I will do the intersection. Then, I will do union with an empty set and intersection with an empty set. Just as an illustration of what this is all about. So, you can pause right now. You can answer what will be the result of these operations yourself. And I will just do it myself right now. Okay, let's think about what will be the union between A and B. Union is an operation which results in combining all elements from the elements of the one set and all elements from the other set. So, let's consider what will be the union in this case. Obviously, everything from A should be included, which is wall, I'll put current brackets around, wall, building, planet, and book. Next, we should consider the set B and identify which elements are new and which elements are not really containing already in the A. The wall has already been listed as part of the A set, the book as well. But man and pen were not listed before. So, if I'm unionizing A and B, I have to add man and pen. And that's the result of the union between A and B. Now, the intersection is a set which contains common elements between A and B. Alright, what's the common? Well, wall. Is the wall common? Yes, it is. It's contained in both. Is the building common? Well, no, there is no building in B. So, building is not a common element. Planet, neither. The book is actually the common element. So, the intersection contains only these two. Now, obviously you don't have to go through these elements to find common, because you have already gone through these guys and chose only those which are common. Obviously, to go again through the B doesn't really make any sense, because you will not find anything new, which is common. So, it's practically enough to go through elements of one set and pick only those which the second one contains. Incidentally, if we reverse the sequence, let's say we do it the other way around, we start from B and we go through the elements of B and we'll find which are common with A. We should come up with exactly the same result, right? Because if you remember, these operations are all commutative. Okay, let's do that. Let's start with B and do the intersection with A. Both. Is it common? Yes, it is, because A also contains it. Book. Is it common? Yes, because A contains it as well. Man and pen are not common because A does not contain them. So, we came up with the same result. It basically illustrates that A and B or B and A are all interchangeable in set operations. It's all commutative. Okay, great. Now, these are, well, they might look a little strange, but what it means, let's just think about this. We are unionizing A with an empty set, which means we are combining elements for one set and another set. So, all elements of A obviously will be included, which are wall, building, planet. Okay, and we have to add all elements of the second set, but second set contains nothing. It's an empty element, empty set. There are no elements there, so there is nothing to add. That's it. So, the result of the unionizing between A and empty set is basically the same thing as A. All right. How about intersection? If we are intersecting a set with an empty set, well, what intersection means? Common elements. If this guy does not contain anything at all, there is nothing in common. So, no matter how big A will be, when you are intersecting with empty set, you will get exactly empty set, all this. So, these are very elementary properties of the set operations, which I wanted to demonstrate in this particular exercise. That's it for now. Thank you very much.