## Transcripts

1. About the course: higher ready before starting this course. I want that few more points so that you can get the maximum out of this course. The first important point is have a very good notes. Okay, While watching the lectures, please make some good shark notes so that you will be able to read in the concepts. After some few days seen Budo, you are able to understand the concepts, all the concepts very well. What might happen is after three or four months, there is a very high chance that you might start forgetting the concepts. So the best way to read in the concepts for a long time is to is to make shark notes. So I would highly recommend you to please make short notes so that the concepts can be reading for a longer, longer time. The second most important point based X books. See, though everything is covered in this course, I have covered everything that is possible in sectarian algebra, all the concepts. But still it is very good to go for some expert that export, which I am using, is in a Thatcherism Discrete mathematics came by can attach chosen. It is a very good textbook. The concepts are explained so well in the export. In case if you want to go for the textbook, they can go for it. See, these are mice traditions. Okay, In case if you have a better idea if in case if you have a better plan, you can always go for it. I'm perfectly fine with this. Okay? In case if you find better textbook, you can go for it. No, the third most important point is this course actually solves in this course. I have saw a lot of problems. Okay? A lot of problems. These problems are. These problems are great problems. It has been taken from from some world class in your stories, like a mighty Stanford and all this. I have taken those problems from those websites so that it actually increases every infusion. Your logical thinking. Okay, So what they want you to do is don't just watch the lectures alone. Also prayed to practice The problems by us sells first. In case if you're not able to do it after that, you can watch the lectures. What way am telling this is in case if you solved the problems by your own what you might be able to understand what very were actually lagging in terms of concepts. OK, so these are my three. These are the three points which I want that before starting the course so that you can utilize the course the best way possible. So that's it from my side. You can start watching the lectures. I'm sure that you will have a wonderful learning experience. I have put all my energy in making the schools. I'm sure I'm very, very sure that you will have a full class learning experience in this course. Okay. Thank you.
2. Introduction to Sets: before seeing that wants concepts of Secretary like lattices partial our house and all this . Let's first have some quick overview of basic concepts. Okay, Very, very basics. These are very basic concepts of Secretary. I mean, I'm going to start with Waters when Bay set, and then I'm going toe go through very, very basics. Okay? Now, in case I know, most of them will be comfortable with these concepts. In case if you feel like this going, you know, you are aware of these concepts, you can skip this section and you can proceed with section toe. OK, These concepts I'm covering for people who are not very much aware with the basics. Okay, because they want to start from scratch. Fine. Now let's see what this one Base it again. Hey, Said is nothing but a collection off another besting objects. Okay, It is a collection off are not a distinct objects. Generally, it is represented using two braces. Okay, a sit is represented using to brace is on. We right. The elements there there's have separated big comma. Okay. For example, this address three elements want to 13 and they're separated by a comma. Similarly, this. It has two elements. Okay, so this is what we mean. May say why we have used our northern Paris in Gezi by right. You could do 123 are you could do 231 It doesn't make any difference. Okay, I can I write it this way are right. It is this way. It is not actually indicating the harder. Okay. Does not say it is just saying that Yes. Is it which has three elements? 12 and three. It doesn't say that one is the first element to is the second element or three is the third element. Nothing like that. It is not actually specifying the order. It was just saying a set off elements. Okay, fine. Similarly be a suspect which has two elements A and B are You can also say that as being you. Okay? Nothing wrong in it thing? No, they have used distinct here because And guess if I have you equal, do one comma one come out to We don't rate it this paper again. Rather related as equal to one. Come on, a good one, Comotto which means we don't rate the duplicates. Okay, insets we don't specify duplicates. Okay, That is what we mean by sexy. That is also sets where we can specify duplicates. We also say there s multi sick. Okay, it is a very advanced concepts of concept of Secretary. Actually, this is not there nor cause what I wanted list throughout this course. But not I use the word said it means that only distinct elements are love. Engaged by how one comma one Come on toe. I can simply write it as one comer toe. Okay, this is not alone because it has duplicates. Fine. No, there is a kind of said called Esnal. Sit. Okay, now, this said has three elements. Okay, this sick has three elements. So we say that car minority off this set a history. Okay, Cardinal, the office set is nothing but a number of elements which are presenting. Say this it Okay, in case if you have a set, you called the number of elements. And then they say that is the carnality of the sick. For example, coordinator of this at this too. So what? This one final status? Yes. Set. Whose Cardinal D zero we say the decimal said Okay, you sit with no elements and see in rather than saying is cardinal decor 20 I can also say that they said with no elements are except with zero elements is called Arsenal said right ? No, it is denoted by a will do embrace with no element inside it. Okay, we have to brace is now in case of this set here some elements in it. You'll right inside it right now. Here we have. We didn't write in the element inside it. It means that he designed all set. OK, This sect has no elements in it. We can also present this as a quarto by Okay, we can also present This is equal to pay, which is perfectly fine but important bind. You need to notice You should not write it as you could do. You could do braces inside which he cannot write. I forget this is not a low. These two are fine and they indicate empty set. But this doesn't This is this is a valid sec. Definitely This is a valid set but it is not Announce it okayed Indicate something else. We will see the bit later. Okay. What? This actually indicates he will see it a bit later. But as off No, Understand that. And I said this represented by either this art this but not this. Okay, this is wrong. Fine. So again, this sex can be classified into various types. OK, they are called us finance. It's under another bonus in finance it. Okay. For example, this set A and B are fine. Eight sets because it has fine. Eight are accountable. Number of elements. Okay, because it has three elements. It there's two elements. Whereas if you take the Sadako even natural numbers, it is an infinite sick. Great, because it has infinite number of elements. Okay, fan, No. Let's see some more basic concepts. Okay, this is this is a very basic introduction to sets now in games. If you feel like it is very, very basic, you can always keep the section okay, because this section, I think most of them would be a benefit. But still, I want to satisfy every student that same covering these concepts. Okay, fine.
3. Subset, Superset, Proper subset, Proper superset: No. Let's see what is one base upset and properties upset. OK, see, subset and proper subset are actually used. The compact, Any two sets. Okay, in case it there given to sits here on B, we can compare these two sets using these two concepts Name that's upset and proper subset , for example. You can say that is a subset off. Be you friend would leave every element off year is presenting. Be okay. See for example Hey Gordo Oneto OK, I'm being called. Oh, 123 Here is a subset off me because every element often is present in B. For example, one tour presenting be rate. So Yaser subset off me. No Weatherby subset of here or not, it is no right. It is not as that's upset way because threes presenting be But it is not presenting here. So be is not a subset off you. Okay on it is denoted by a subset me. Which means if he is a subset of b, we can be noted like this. Okay, fine. That is This is not well. This is not valid. Curate. See, here he is upset of B But B is not a subset offi coming toe this example. Okay, fine. Now what? This one big proper subsidies first thing is in order to become a proper subset, it should be a subset. Which means if you A is the proper subset of be in case. If you wanted to say like this, then these two conditions should be satisfied. Okay, Which means every element off year is present in B for example in order to in order to say that is a proper subset of B every element of years presenting, being which is valid here also be has we should have at least one Ellemann which year doesn't have Yes, bees Having one element with you doesn't have so he is a proper subset of be ok in Greece. If he wanted to say that he's a proper subset of B In that case, every element off you should be presenting beat as for less be should have at least one extra element Which year doesn't have okay here Yea says is a subset off b has less is a proper subset of be okay. No, Michael Michael itchiness Whether bees a proper subset of year not CB is not a subset of EA itself the now getting the proper subset rate. See, first conditioner cough proper subsidies Actually indicating subsidiary in order to say that is a proper subset of three first year should be a subset off be. That is what first condition is indicating after that. There's also having one more condition. Okay, so if he is not a subset of B, then it cannot be a proper subset of me. OK, that is one important point. No, let's see some examples. Okay. For example, a will do 12 be called one. No is Yes. Upset of beer not is not a subset of B because every element of years not present in B But these is upset off year as well as being the proper subset offy way. Because every element of B is presenting here. As for less, he has one extra element which B doesn't have. Okay, So bees upset up? Yes. Will it be the proper subset off here? No. Let's take this ce called. Um is it okay now? Is CS upset? Appear not? Yes. See is a subset appear because every element that C is present in the a. C. C is not having any element rating gives it some 11 is present. That element should be presenting years Will now see is not having any element on DSO Seizes up Qaddafi him so 11 important point you needed notice is just very, very important is um he said BBs ups, adults Any other sit again? You take any set MD said will be a subset off it, which means every element of this said surely presenting any other set. Okay, fine, so fine. No, My question is whether sees a proper subset of year. Yes, it is, too. Why? Because every element of sees presenting year as for less years having at least one element which is not present in sea similarly be is a prophecy is a prophet subset of bs Feel right fine seduce what we mean by upsetting proper subset. Okay, See here doing bottom binds. You need to notice if you is a proper subset of B and definitely use this upset off me. Why is this true? If he is a proper subset of being means that these two conditions are satisfied, right? If this is true, this has to be proved. That is what it doesn't get. It is indicating now this is Prue. These two point should be true, but is as well they're saying that first point is true rate. If two points are true, then first point This crew on the first point is true. We can say that this is true. This is valley. Okay, fine. No, it is not a proper subset off be then says idea is not a subset of B then definitely is not a proper subset of B, which means you first condition is not said this faith, it is as good as saying that these two conditions are not satisfied. Right? One leave for so if he a is the proper TV is not a proper subset of even a is not a subset of be than definitely is not a proper subsidence be okay, that is one of these two points are very trivial, right? It is very difficult to explain. I think usually should be able to get it OK, by considering these two statements Now, let's see what is meant by a Super seven proper so percent. Okay, it is the same concept. It's a played here. As for okay, now, in case you feei is a subset off B, which means if you every element of the age but sending B we can we can we can actually the real to conclusions from this. Okay? One thing is he is upset off be. We know that we can also say this is B is a super set off. Okay, He is a subset off b means b is the super set of you know what is one big for a proper supersede again? The same is valid. If he is a proper subset of B, then we can say that b is a proper super set off e. Okay, In simple words, I can say that B is a proper super set off year. You found one. Leave every element of years presenting be as well s. We should have at least one element, but he doesn't have okay that that that this When we say that Visa proper superstar Tuffy Okay, one more important point. If he is the proper superstar proper subset of B then B is a proper super set off here. Okay, I think