Please, Translate the following English into Korean: 
Okay, so let's start. So thank you, thank you for coming my class, our class, or anyway. So this class, the title of this class is Probability and Random Process, SW3026. And my name is Youngbuk, and then, so today's lecture is the first, no first, not the second week lecture time. So, okay, so let's begin. Okay, so now, let's, before we jump into the probabilities or random process or whatever, so then, let's think about the natural phenomena. So there are so many natural phenomena, such as signals, or the movements, or, how can I say, the text itself is actually one of the natural phenomena cards. So even that one is written by the human. So the human itself is also the natural element, or something like that, so then, as I said, the movement itself is also kind of the natural phenomena. So that's why it is easy to say that, so the natural, the text itself is natural phenomena. But anyway, so what I want to say is that there are so many phenomena in nature, but we can classify those phenomena into two parts. The first one is a deterministic phenomena, and the second one is random phenomena. So then, what is the deterministic phenomena? So the definition of this deterministic phenomena is that observe the same result when the conditions are the same. And then, but otherwise, but the random phenomena, there is observe the different results, even the conditions are the same. And then here, we can say that almost, almost the natural phenomena, the phenomenons are the random phenomena. Since, well, let's think about the coin toss, which is a quite simple one. So we just plip the coin, and see the result. The result itself is the face of the coin, right? So the head, the arcade. But here, when we coin the toss, then can we determine the result of the coin? Head, arcade? It's hard to say that. We always said that one has random. So that one is actual random, but we expect that the probability of the result, I mean the head result of the coin is 0.5, and then the other one is also 0.5. If that one is, the coin itself is fair, something like that. So what I want to say is that almost natural phenomenons are the random phenomena, then we needed to model these randomness for doing, for describing, not only describing, but modeling the natural phenomenons. That's why we need to learn about the randomness. All right, then what kind of mathematical models for handling this randomness? So, but there are so many things, but I want to say is that the procedure of the mathematical model looks like this. First, we need to build the models, which is identification of the model, so we needed to build an appropriate model for the natural phenomena. And then we should solve for certain, so we should suggest a solution for a certain quantity of interest. So then that's the actual mathematical way. And then after building the model and then the solution of the model, then we need to verify the model by doing the physical experiments. And then after the experiments, then we need to modify the model, so based on the experimental result. Then, all right, and then without these mathematical models, we can apply to similar other situations and prediction, predicting the outcomes. And then we can suggest alternative solutions for a given problem, something like that. All right, but, well, maybe you might say that, you might say, I understand it, but I don't know exactly the meaning of this one, so let's talk about the one example. That's the coin toss, which I mentioned earlier. But anyway, so the coin toss is, actually a coin has two bases, head and tail, right? And then when we flip the coin, then the result of this experiment, kind of experiment, right? So, complete the coin, the result of the experiment is head or tail, right? Then we know that, so probably that a coin toss will come up heads is 50%, if the coin is fair, right? And then the coin will come up head about half of the time if we flip the coin or lastly negative, so we will, actually we will discuss, we will see this kind of statements. But anyway, what I want to say is that thinking about the coin toss, then let's move on to these mathematical procedures. So the identification model, can you see the slide? Maybe I needed to do it by pen, all right? So the identification of a model, think about this one. But before we do that, sorry, so before we do this, let's think about the coin toss, like I said. So then the coin toss means that we need to, we wanted to know the result of the flipping the coin. Then the result is two things. It is the first time we use this one, sorry. Maybe there are some mistakes, but anyway. The result of the coin and the result of the flipping coin, the experiment is the head and then the tail, right? Then there are two types, but we can't, we wanted to know the result, head one, for the next, the flipping the coin. Then we can say that, so the probability of the head, we will check about this notation in a little bit. Anyway, so this one is a small p. Or just less, I think small p is good. So small p, then that's the building a model. So that's all. So building a verification model means that, sorry. So verification model is that, so understanding the problems and then build a kind of the structure of the mathematical model at the first time. And then the next step is a solution for a certain quantity of interest, right? So then we can say that, so the probability of the head for flipping a coin is p and then that one is computed, maybe, not maybe, I'm sorry. We can assume that this one is computed by the number of the all flipping. And this one is the number of the head. That means that, so we can say that the probability of the head is the number of head over the number of the all flipping trials. So then that's the solution for a certain quantity of interest by mathematically. So intuitively, mathematically, whatever. So we can set up this kind of the formula for solving the problem. And then the next one is the verification of a model, right? So that's the basic experiment. So we might do the flipping the coin a lot, maybe for example, a thousand, wow, that's hard for me. So the 100, whatever. And then the observed the results. So that means, so for example, as I said, so the number of the trials is 100 and then we know that, so there are 51 trials, I mean 51 coins shows the head at the result. Okay, then that's the verification model. And then the last one is a modification of a model based on the experiment. Actually that's the one. So with this result, then we can set this probability to be 0.51, something like that. So that's the procedure of the mathematical models for the natural phenomena. So is it, that's all. And then so I said that, here, okay. Then use this set of mathematical model is that we can apply to similar other situations such as rolling a dice or other things. Maybe you're rolling a dice because the dice has six faces, right? But coin only has two faces. So from this coin toss experiment or coin toss modeling, then we can extend the model for the more complex natural phenomena which is rolling a dice. And then we know this experiment and then modeling, then we can also predict the outcome of the new things. And then we can also suggest the alternative solution for a given problem where we did this. And this example is hard to say there, but if we solve another kind of natural phenomenon, then maybe that one is actually quite complex problem. Then means that we can also suggest various solutions for solving this problem. So if we, that means that if we build the one mathematical model, then from this kind of the motivations or ideas or observations or experiments, no, no, experience, then you can build another solution for solving the problem. So that's why the mathematical modeling itself is very important for understanding the natural phenomena. All right, okay, oh yeah, yeah, then. So what kind of the complex natural phenomena or the problems with the probability? So as I said, I'm a natural energy processing researcher, so I wanted to introduce one of the big problem in NIP area, so that's the language model. So the language model itself is a kind of this one. So this one actually predicting the next word for a given sentence like that. Well, yeah, and then the language model itself is actually used in various natural language processing applications, such as search engine. So actually this figure is a screenshot of the Google search, and when I type the Y people, then Google suggests a next, sorry, how can I delete it? How can I delete it? Oh, yeah, good, okay, and then depend. Oh, no, sorry, and then depend, uh-huh. So, sorry, so anyway, so when I type the two words, Y people, then Google suggests that these kind of the, next words, right? So for example, the Y people smoke, Y people drink liquor, Y people cheat, and Y people don't wear masks, something like that. I think that's the new things, because nowadays they're copied in 19. But anyway, so that means that, I'm sorry, and this kind of problem is called keyword expansion, since Google wants to predict the user's intention, why the user wants to search something, and then when Google take a part of the input, then he or she, or just Google, wants to minimize the size of the problems, and then predicting the intention of the users. And then they also wanted to help to make good queries to find the relevant documents. So that's the actual duty of the, duty or the functionality of the search engine. So that's why Google also built this kind of language model, which I said, so the language model is actually, as I said, that's the, they don't actually predict the next word. And given the previous one. Like this one. So then here, why do we need to think about the probability? So can we know that's the Y people smoke? That's quite appropriate phrase, appropriate sentence. Not sentence, but it's more appropriate sequence of words. However, we can also know that's Y people, me. Can you think that this, of these sequence of words is appropriate? It's not actually, no, right? So what I want to say is now, so given the Y people words, then we know that these words are appropriate, and then these words is not appropriate for a given words. Then how can we model in this kind of the problems? Then we know that's the probability of the smoke, given the Y people, it may be 0.2, one something like that, but probability me, given Y people, is 0.000001, something like that. So if the language model knows this kind of the probability, then it can give a good result for keyword expansion. So that's the functionality of the language model, oh yeah, sorry, language model, and then the Google actually wants to build a good language model. So for the students, maybe someone heard about the terminology, but no, so this class is not a natural language processing or the AI class, so that this is the probability class, so I just skip the details of the language model. Okay, so then let's move on to the textbook part. So before we know about the probability and the random process, so let's review the mathematical background, such as the set theory. So in this video, so we are talking about the, I will talk about the set theory. And then maybe you already know about these kind of things, so you can skip if you know everything. All right, so let's begin. So the A set is an unordered collection of things, and then we set these things as elements. So for example, here there are four sets, A, B, C, and D, and then A, the element of A, elements of A is, what's that, the clover and the diamond. Then we note these kind of things, means that the diamond, which is the element, in the A is a set, so this is A, the in, diamond, the diamond is in a set A, right? And then we know that the heart is not in a set of A, right? That's why, so we draw these kind of lines in the, between this, now how can I say, this figure, something like that, I'm sorry. All right, and then the next one is B, one, two, three, that's quite simple, and then the C is, we can also note the elements, or the set is like that, so the X square, then dash, and then the X is actually one, two, three, so that's the same as this kind of the rotation. So that's another way to represent the set, and then the, what's that, the element of the set is, the type of the element of the set is unlimited, so we can create these kind of characters as an element of the set, so that's actually the head and tail, like that. All right, and then there are so many sets, as I said, then actually we can create the natural number from starting the one, two, three, four, something like that, so that one can be treated as a set. And with the same procedure, so we can also treat the integers as the set, as a set like that, and then the real number is set is also a set. Okay, and then the next one is the subset, okay, so then we know the, what's that relationship between the set, so here the set A is a subset of set B, if every element of set A is also an element of set B, then we write these kind of notations, and then here we can denote that this indicates the subset. So more formally, so we can define this subset, it's like that, so the all, so here, so the all X, which is element of set A, then we know that this X is also in the set B. So for example, so the E is, there are two elements of the set E, one and four, and then C has three elements, one, four, nine, and we denote the relation between the E and C is like that, so we know that the set E is a subset of a set A, as a set C, like that. And furthermore, so we know that the natural number is a subset of the integer, right, because, you know, so here, so integer is this kind of the things, and then the natural number is here, so then, but these part are actually the natural number. So we can say that the natural number is a subset of the integer, like that. Okay, and then the next one is this one, so A equal to B, and then if and only if the A is a subset of B, and the B is a subset of A, so this kind of things. So here, that's one of the important properties of the set, so the order of the elements in a set is not fixed. So that means that one, two, three, and three, two, one, those are actually the same. Yeah, but anyway, so then here we wanted to say this, so when we say those two, if the two sets are the identical, then means that between them, one of them is another, subset of another of them, vice versa. Okay, and then, yeah, the next two terminologies, the universal set, so that's actually important. So the definition of the universal set is the whole thing. So more precisely, the set of all things that we could possibly consider in a given context. So here is important, a given context. So depending on the context, the universal set is actually changed. So we denote this universal set as S over omega, and then there is the opposite concept, named null set. So null set is actually, there are no, no, it's hard to say, no elements, but anyway, so no elements in the set, yeah. So one of the interesting properties of the null set is that for any set of A, null set is a subset of any set. So that's also quite important. And then we will use this kind of, these properties in this class more and more. All right, and then next one is the band diagram. So that's also, that's actually a tool to describe the sets. And then it's very useful, so I will just review this band diagram for you. So in a band diagram, so any set is depicted by the closed surface like that. So the square is a universal set, and then the set A is denoted as the sucker. And then there are several operations among the sets. The first one is union. So the union of the two sets A and B is denoted by this notation. And then it consists of all objects in A or B. Yeah, here, so the or, that's important, right? A or B. So that means that the X is in A or X is in B. Yeah, or that's important. So that's one of the example. And then when we use the band diagram to denote this union, then we can denote these unions like that. These are the A part, the green one, and then these blue ones are B part. And then we know that the union of the A and B is the all of these areas, actually. Not this part. Okay. And then the next operation is the intersection. So the intersection between two sets A and B is denoted by this part, a cap. Yeah, right, sorry. So in English, so the name takes out of this one, it's a cup, no, the cup. And then this one is a cap. Anyway, so the A, cap B, and then consists of all objects in both A and B, so here, the end. So that's important. So when we compute the intersection, then those elements, I mean, actually, what those areas are actually the intersection between the A and B. And then next one is the complement. The complement of set A denotes the A complement. So the C here is the set of all elements in S that are not in A. So what it means, so you know the S is a universal set, so the all elements. Then the A is this area. Then we can say that the complement of A and C is the other areas except the A. So some field use this notation to denote the complement of A. And then, okay, so the next part is the difference, difference or subtraction, so something like that. So the subtraction of set B from A, here, the order is important, so B from A. So here, the A minus B. That's one of the problem in English. But anyway, so in English, so the subtraction of set B from A means the A minus B. It's all elements in an A that are not in B. So here, sorry, so the A minus B means that those kind of things. What's this one? So it's easy. So means that A minus B means that there are A's in here. Can you see the co-stars? But anyway, so the A is here and then B is here. So we remove these A intersection B part. And only remains those part which are not in B. Then we can also denote this A minus B means that A cap B complement. It's easy to do that, so let's skip. All right, okay, then there is also important things, the mutually exclusive set, the disjoint, especially for understanding the probability. So here, the two sets A and B are mutually exclusive, mutually exclusive or disjoint. That means that there are no elements between A and B. So means that the A cap B, so intersection between A and B is actually null set. I said, it's a null set. So then, A1, A2, then An are the mutually exclusive, if all the elements, all the elements, not all the sets, sorry, so the intersection between all sets are null set. So we can denote this kind of mutually exclusive sets like that, there are no intersect areas between the sets. Okay, so and then we can expand these concepts. So the partition, I thought this was important. So the partition is a collection of sets. Here, so the set, right, but we denote the set of set is actually just use these one collection. Because there is some little difference between the collection and set, but I don't want to describe it too much. One is what's the difference, but anyway, so think about just where correction. So there are several sets. Or, you know, so you are the software engineering, no, no, software engineering, software, anyway, software, you know, the major students, then we can treat the set as a list. Oh, no, no, no, the correction is a treated as a list. So anyway, so the collection of sets, A1 to An is a partition of the S. So here, this is important, two things, the partition and then of what universal. If they are joint, it's joint, as I said, it's mutually exclusive. And then here, the union of the all sets is the S, which is a universal set. So when those two properties are matched, then we can say that these collections are the partition. Okay, and then the next one is the De Morgan laws, De Morgan and Bobchick in Korea, but anyway. So that's one of the famous ones. So the A cup B, which is a union of the A and B's complement is the A complement cap B complement. So it's, well, how can I appropriate that? It's easy using the band diagram, but because of time, so I don't want to say about it, and then you already know about this kind of things. All right, and then the example part, yeah, you can solve it by yourself, it's easy, right? So the S, even the universal set is six elements, and then A is one, two, and B is two, four, six, and then you can compute all the things. That's easy. And then, oh yeah, then the next one is the distributed law. So that's also important theorem. So the A cup, patterns is B, cap C, it's the same as the A cup B, and the A cup C, the intersection between the A cup B and then A cup C, something like that. And then when we change this one, then this one is also follows to the order, and then that is called as distributed law. Okay, and then the next one is a function, so that's very important, so please remind, or leave you whatever, and truly understand this concept, function. Because actually the random variable, we will discuss about it later. So the random variable itself is, if even this random variable has a name variable, however, it is actually the function. I will talk about it later, but anyway, so now let's just talk about the function itself. So the function is denote the F, and then there is two things, X and Y, and then this arrow is actually the function. So then here we can see this one in, I'm sorry, we can see it on the figure, so the domain, and then call domain, and the row names, whatever, but from one of the elements from the domain, then when we use in the F, then this element goes to the, these area, which is called the names. And then what about the whole domain? Actually, so that's the definition of the function, so from X to Y, then X is a domain, and Y is a call domain, then, okay, so then maybe the question is changed. What is the range? So the range is the set of all possible value of Fx. What it means, so the old elements in the X go to this range part, but we can make it define the call domain, which is a little larger than the range, so that means that the range is a subset of the call domain, right? Okay, so let's think about the function with example. So here we consider the function F, the R goes to R, so R is a real number. Then we define the F function is like that, X square, so Fx is an X square. Then we know that the X and Y is actually same as real number, but the range of the F is actually positive real number. Cause you know that it is a square, right? So when we draw the graph, we already know that, so this is the X, okay. So X and then Y and then this one, right? This is the square of the X, right? And we know that these lines are not in this part. So we can say that, so X and Y is a real number because that one is the definition of the, when we define the F function, then we set this like that, however, when we think about the range, then that's the positive real number. And here, one of the important concept here is the one to one, one invertible. So that means that if the Fx is equal to the Fx2, then X1 and X2 is the same. Think about the, for example. We will discuss that very later with a real problem. I mean, maybe next week, next week or whatever. Okay, then let's move on to the countable and the uncantable set. It's very interesting. So here the concept of the one thing in the cell is a cardinality. So the cardinality of ZA is the number of elements in A, and then we can denote it like that. So that's easy, right? But why people say about this one is as a cardinality, because of the infinity. Okay, so let's move on, let's see about it. So the ZA is finite if the cardinality of the A is less than the infinity. That's the definition, very interesting. But then ZA, if countable, if it is finite or the element of A can be enumerated or listed in the sequence A1, A2, A3, something like that, and then that's the definition. That's weird, but anyway, so let's talk about it first. There's a finite, for example, one, okay. So one, two, three, there are only three elements. Then here maybe we can say that this one is B. Then we know that the cardinality of B is three because there are three elements. Then we can say that the B is actually countable. We can say, we can count one, two, three, right? That's count, all right? Or here, one of the interesting is that if the number of elements is infinite, but we can also say that this one is countable. Then how? So the AK, so there are so many elements, A1, A2, whatever, A1, A2, A3, whatever, whatever. And then the number of the elements is infinite. This is infinite, however, we can say that this is countable. So for example, it's actually the natural number is countable, so we can count one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14. This is right? We know that, so natural number is infinite. So we count these numbers. Infinitely, however, I can say this, it can be count. Yeah, even this one is infinite. Then what's the another one? So that's the uncountable, so not countable. So for example, for example, the real number, or real number between zero and one. So is this not even in the other part? Oh yeah, but anyway, yeah. So here, so let's move on to here. So this notation of this one is like that. So there is a X, which is in a real number, and then the range of the X is between the A and B, and the A and B actually are in this set. However, when we use this path with this, then the B is not in. Okay. Okay. Sorry. B is not, B is not in the set. But anyway, so here, as I said, so that's the, maybe zero and one. So real number between zero and one. You know this is too small, right? It's one. Length of the, as I said, that length, if I said length of this one, then we can say this is one. You know, from one circle to one. However, when we think about the real number, then it's the cardinality of this set is uncountable. So a set is accountably infinite if it is in one to one corresponding with this. Hmm? So accountably infinite. So that's one of the other things. So for example, so jet, which is a set of integer, is countable, or uncountably infinite. So that's one of the interesting part. So as I said, so the natural number is countable, right? However, and then furthermore, the integer is countable. It's also countable in terms of the cardinality. So that's very interesting. You know, as I said, so n is actually subset of the jet. You know, the natural number is a subset of the integer. However, when we think about the countability, then n and jet is actually same. How? So we can order the integers like that, starting with the zero, starting with the zero, and the one minus one, two minus two, three minus three, four minus four, something like that. Then we can map, as I said, here, so one to one corresponding, so one to one. So we can make a F function like that like that, the zero goes to zero, one goes to one, two goes to minus one, three goes to two, four goes to minus two, something like that. And as I said, no, that is cost infinite, this one is also countable, sorry, in this one also, this one can also go to infinite, but the sequence is countable, and the one to one matching, they want all matching, so that we can say that the set of integer is countable. Okay, yeah, so we can check out value later, and then one of the interesting is the positive relational number in Korean, 유리수, which is also countable set. Okay, that's one, it means that you can find the function from natural number to this positive relational number. Okay, made. And then, yeah, okay, and then the, but here, the real number, that's not countable. So in fact, any intervals A and B, where B is greater than A is not countable, means that, you know, that any interval means that so, an interval itself is a subset of the real number. Then, but how can we prove this one? So that's very intuitive, and then actually this one is just by the count, and actually I love it. I love this, wow, I'm very surprised. Make this simple but powerful proof point. So you know, this is not a mathematical class, but anyway, I will just briefly explain about it. So when we use, when we, so we will use the proof by contradiction to prove this statement. So we assume that, so the f n is equal to r, so the, that means that, so the r is a real number, and n is the natural number, like that. So that means, so as I said, so, if we wanted to show that the one set is a countable, then we can make a map, one to one map, from the natural number. So here, I just make a one, one to one, the one function from natural number to real number, and then, because for the simple part, so and then I said, so any interval between a and b is not countable, so we just set the range between zero and one, okay? And then the f is looks like this. So then instead of f one, is zero one, one, two, two, three, four, five, six, eight, something like that. So it goes infinity, right? And then you know, then these numbers can also go to the infinity, right? That's the real number, right? And then we can, then from this map, ping, we can make a new number x, like that. How? So, except this, no, this is the first one, so we take the first number from this f one, and then add it to one. So the one, this one, and the plus one, then here goes to two, and then we pick the next one from the f two, then we add the one, then it goes to four, right? The same procedure, six, nine, three, so we can pick these kind of the n's number from the f n. So then, as I said, f and maybe n is zero point, something, something, something, something, but here, this one is the n's number, right? And then we extract this number and the plus one, and then if the number is nine, then just go change plus one, if you know the 10, then just take last number, zero. Then, with this procedure, then we make this, a new number, zero point two, four, seven, four, something like that. Then, I said it's a new number, because maybe use the, because it is created by the new method, right? Cause, you know, even if this is n-family, but let's think of, as I said, maybe let's think of the t's one, then maybe, then we, maybe, and then the t is f is a zero, two, four, seven, four, something, something, something, right? However, even all other things are same as the f x, however, the t's number is actually different, right? Because of this rule, right? For example, this number is the one, then x might be, this part is the two. Then we can say that the f t and then x are not the same. But however, x is f is a real number between the general and one, so then f is not the one, two, one function, right? So that means that there is no f, which is a one, two, one function from the natural number to real number. So that's why the real number is not countable. That's simple, quite simple, yeah? So here I wanted to say this, that's it, yeah, right? So anyway, as I said, this is not a mathematical class, but I wanted to say the review of these sets, the operations and then the functions and then the countable and uncountable. This one is actually used in the next, or next next week, like that. Okay, so thank you, thank you for listening my presentation. Thank you for listening the class. Okay, bye.