 I'm Zor. Welcome to Unizor notification. Today's topics include the concept of sets. Well, there is a well-defined mathematical sets theory, which we're not really going into any depth today. I would just like to explain certain concepts of sets theory, just to be able to use these concepts in further study of certain subjects, which we will touch. Set theory is really very deep, and it includes lots of different topics. I will just touch some of them, and I will try to exemplify the concepts which are involved. First of all, what is a set? We don't really define this. The concept of a set and element of this set are so elementary and probably the best would be just to explain and just put a couple of examples of what it is. You can consider a set of points on the plane. You can consider a set of people who live on the planet Earth, a set of triangles which you can draw on a piece of paper, whatever. Any object can be an element of some set. Without a rigid definition of the concept of a set, we can probably understand that you can introduce a concept of empty set, the one which does not contain any elements, and in mathematics it is usually simplified as this straight zero or something, whatever. So there is a concept of a subset. Subset, if you have certain set which I can just exemplify as, for instance, set of all the points in this plane which belong to this closed object, all these points are set. Only those which belong to this closed contour can be considered as a subset. So we have a set, we have a subset, we have an empty set. Now a few operations which you can do based on set theory. Again, the only thing which we are talking right now about is whether element belongs to a set or does not belong to a set. So there are certain operations on sets which you can imagine. Like, for instance, if you take all people who were born prior to the year 2000 and people who were born from 2000, 2001, so one is less than 2000, greater than 2000, and less than 2001. So this is one set and this is another set. Well, obviously, if you consider people who were born before 2000 and from 2000 but before 2001, you think and quite right that you will get the set of all people who were born before 2001. So we have made some kind of an operation from two different sets we have created the new one. So you can perform certain operations with sets and they are pretty well defined in mathematics. So I'll just put them on the board. It's very simple and very quick. If you have two elements, if you have two sets A and B, you can perform the operation of union. Out of two different sets we have created a new set which contains all elements of the first element and the second element. So basically, if you have, let's say, this as all elements of the A are on this plane inside this contour and then you have this as B. As you see we have some common elements as well. So all elements which belong to either of these two are basically here. I will just put my new contour here. So all these elements belong to basically a union of A or B. Logically speaking, we very often use the word or because how to describe these elements, these are elements which belong either to A or to B. That's what union is. Fine, next operation. Next operation is called intersection. What's the intersection? Again, let's use these diagrams which by the way are called band diagrams. So if this is A and this is B, intersection as pretty obvious from the word itself is something which belongs to both. Only this area which belongs to both is an intersection, the definition of an intersection. And very often we will use the word end to describe intersection because these are elements which belong to A and B to both of those guys. Well, just as an example, what's an intersection this and this if there are no common elements? Well, intersection between these two guys is empty set since there is no elements which belong to both of them. So this is just one of the small properties of this. All right, we've got union and we've got intersection. Another concept, concept of a subset. So if you have a big set, let's call it U, a reversal set, and some subset of this, let's call it A. There is a concept of a complement, a complement of A towards the universal set U which A belongs to. Basically, the complement is everything which does not belong to A. So it's this area around A. So all these elements which do not belong to A are complement. And with a complement the word not actually is associated very much because these are elements which do not belong to A. Now, a few very interesting rules in the sets here in which you might find kind of common arithmetic. Basically my point is that union is very much like addition and intersection is very much like multiplication. Well, obviously I cannot say that this is this but there are certain similarities between these pointages, between these operations. So first what's very important is these operations are associative and cumulative and commutative, sorry, commutative. So the associative rule is this. If you have three different sets which you are trying to unionize together, you can do it in this sequence, first A and B, sorry, A union B and then union C. Or you can do it in this sequence. Well, associative law is basically this. Commutative law is saying that the sequence is not important. By the way, the commutative law is not always kind of obvious. There are certain operations which are not commutative but in theory which we are talking about among sets, the operation of union is commutative. Now, absolutely the same thing is with intersection. So if I will just replace the union with intersection both associative and cumulative law will hold. So intersection is associative and commutative. Great. Now, why am I saying that there is certain similarity between addition, let's say, and union? Well, look at this property. If you take any set and unionize it with empty set, well, by definition of union we will get another set with elements which belong to either A or empty set. Well, nothing belong to empty set so basically only A remains. What's the similarity with addition? Well, if you take some number and add zero you will get the same number, right? There are other similarities but this is kind of obvious. Now similarly, if you do an intersection with empty set well, look at this this way. We need a certain set of elements which belong to both A and empty but nothing belongs to empty so you can't really have anything, any element in the result. So the result will be obviously an empty. By the way, similarity to multiplication is obvious with this, right? Now, there is another very interesting law. It's called distributed law. Among the numbers it's something like this. The parenthesis around so it's more obvious. So if you have a sum of two numbers multiplied by the third one you can multiply first by third and then second by third and add them together. Very similar law exists among sets using union and intersection again in this kind of similarity. So if I will have A union B intersect with C it will be the same as A intersect with C unionized with B intersect with C. Now there is a relatively simple proof of this concept but let me just draw a diagram which will help you to understand what this actually means. So you have a union of two different sets so these are sets A and B. They have some intersection just to make the whole thing prettier and intersect with C. So let's say this is C. Well, now let's think about what is the result of this. First of all, union of A and B, remember it's everything which belongs to either A or B so it's this contour, right? I'll put double line maybe here if it's helping. Now, intersection with C is something which belongs to both this union and C which is this piece, right? So it's this piece of A which belongs to C and piece of B which belongs to C and something in the middle. Well, let's think about what exactly these three pieces can be constructed from. Let's talk about A intersect with B. Well, if this is A and you intersect it with C then you will have this area, right? This one. This is A intersect with C. Now, B intersect with C is obviously this piece and when we unionize it together we get exactly the same thing as in the first place. Alright, so this piece is a distributive law of union versus intersection and similarly, as I was saying, among numbers you have this. Full similarity. That's why I was talking that union is very much like addition and intersection is very much like multiplication. Now, what's interesting and kind of surprising that similar distributive law of multiplication against summation is not really true. If somebody write something like this I just reverse plus and minus as you see here. Well, this is definitely not true. However, what's interesting is that among sets with union and intersection similar formula is actually true. So if I will have this. So instead of union I'm using intersection and this is intersection I'm using union. So this is actually a true thing and again I'm going to just explain it by showing on the diagram. Again, it was kind of unexpected quality after you used to the concept of union being more or less like addition and intersection more or less like multiplication. This is an unexpected quality but anyway, let's just try. So again, this is A, this is B and this is C. A intersection with B is this piece, right? And union with C, it's union with this thing so that would cover this area. So basically tire C plus this piece. All right, let's do it from this side. A unionized with C. Well, this is this thing. A unionized with C, right? Now what's B unionized with C? Well, that's obviously this thing. B unionized with C. Now what's common? Well, obviously the same area which I put all these strikes in is exactly the intersection. Again, let me just show you the game. This is A unionized with C and this is B unionized with C. So what's the common between them, intersection? It's this area, exactly the same one. So this is just an illustration of this property which can definitely be examined in more details. Now, a couple of other properties. Basically, the whole purpose of this exercise is to establish some kind of a language for future studies. So if I will tell something like, let's consider a set of something we understand what we're talking about. Now, another operation was a compliment, if you remember. So if you have some big universal set and this is A, then there is a concept of a compliment which is everything outside of A. Well, let's consider you have two different sets, A and B. And I would like to examine what is a compliment of A union B. So this is A union B, right? And what's the compliment? It's everything which is outside. So A union B and I'm complimenting this. Well, what I'm saying is that it's the same as compliment A intersection with compliment B. That's just another property. Well, let's think about if I take compliment of A is everything outside of A. Everything is outside of A. Compliment of B is everything outside of B. So if I will do something, you know what? I'll just draw another picture and it will be a little picture. So this is my universal set. This is A and this is B. Okay, everything outside of A, I will use these. Okay? Everything outside of B, I will use these lines. So what's the area where I have both lines this way and that way? Well, this is an area as you see exactly outside of both A and B. So the intersections these two areas strike this way and strike that way is all these, all this area where strikes are in both directions. So it's exactly outside of this which is exactly outside of the union, outside of the union. Now, if you can think about the compliment as let's say multiplication by minus one, again using this parallelism between addition and union, multiplication and intersection, you can think that this is from the logical standpoint if you remember associated with the word not like everything which does not belong to A. From the arithmetic parallel it might be associated with multiplication by minus one. And if you do this parallel A plus B and you multiply it by minus one you can obviously say this is minus one times A plus minus one times B. As you see, this is a plus and this is a plus so we don't really change in arithmetic the operation between these two elements which we are adding. In the set theory we are changing union on to the intersection. So that's the difference. The parallel is not really complete. Just sometimes it works, sometimes it's not. This is just an example when this is not exactly the same as in the arithmetic. Now, similarly enough there is a symmetry in the whole thing. The complement of the intersection is union of compliments. Well, again, let's just try to demonstrate it on the diagram. So we have an intersection and its complement. This is our universal set. This is A and this is B. So first we do the intersection and then the complement. Intersection is inside of this area. So the complement is everything outside of it. So let me use these. Everything outside of this area has these lines. Now, fine. We understand that. What if I would like to do this on the right side? Well, let me try to duplicate the picture more or less to my villages of drawing. So first we do the complement of A then the complement of B and then the unionizing. Okay, complement of A is done with everything which is outside of A. So let's use this area which is marked with this direction. Outside of A. All right? Now, everything outside of B. Outside of B we will use this. I'm not that good with drawing. But just for illustration purposes I'm sure it's sufficient. Okay. So now we have to unionize these two areas. Areas strikes this way and areas strikes this way. Unionize means we have to have all the elements which belong to either one or another. So whatever the direction of this strike is, is good enough. So all these points and all these points everywhere there are some lines is our area. Right? And actually everything except this little piece which is exactly the same thing in this particular picture as well. So that illustrates this particular rule. So negation as sometimes this complement called negation because it's not at these two properties. Well, what else is interesting? Well, if you have this universal set and you have a subset which is actually an empty set. So what is a complement of the empty set? Let's call it Z. So the big set is called Z. This is a small one but small empty actually. So what's the complement of empty set? Well, obviously it's the entire Z. Right? Opposite, what's the complement of Z? Well, that's obviously an empty set. Because it's everything which does not belong to Z but there is nothing which does not belong to Z. Z is our universal set, there's nothing more. So we get this. Similarly what else is left? Well, what's interesting about the sets here is that this is a language and not only we can use it for purely mathematical objects like equations, numbers triangles or whatever else we can also use the same type of logic the same type of language with mathematical logic. Now that will be a subject of our next lecture but still what's interesting is that similar notation like union which is equivalent to logical or and intersection which is very similar to logical and or complement which is basically a negation and not. We will use the same language in the next lecture when we will talk about logic. So that's why the language is very important and I would like you maybe to reexamine everything which I was just talking about and again let's just repeat all these little properties which we have properties of associative law for union and intersection, the commutative law for union and intersection distributive law of union against intersection or intersection against union the laws of complement basically and operation was after that. That would be very much like a summary of whatever I was just talking about. So thanks for your attention and we will talk next time about logic and the same language Thank you.