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It's called link tree click it everything will pop up One of the worst cases in UK history David Fuller Okay Talk to me This case takes place in the night at kingdom on the night of the 22nd of June 1987 a 25 year old woman named Wendy Nell was spending the evening with her boyfriend Ian The couple was somewhat serious, but they did not yet live together as the night ended Ian dropped her off at home. They gave each other a kiss said goodbye and off Ian went The next morning Ian received a call from some of Wendy's colleagues They were worried as Wendy had not made it into work and they were unable to contact her Ian immediately made his way over to Wendy's home. Wendy must be a Astute she must have been like one time to work never miss a day type person or a call in and let people know what they do That's like these are awesome. Like good friends to have the moment. She late moment something off But upon entering he made a truly disturbing discovery Someone had made their way into Wendy's home and killed her Ian found his girlfriend lying naked covered in blood Wendy had been savagely beaten and strangled to death and whoever had done this had forced themselves upon her as DNA evidence was found on her body It seemed like someone had been watching Wendy and when she went out with Ian They took the opportunity to creep inside her home through a faulty window and Wait for her to come back on the blouse that Wendy was wearing the police found a shoe print The shoe print likely matched a Clark's sports trek trainer The person who killed her had also taken her diary purse and key ring Most likely taken as a keepsake to remind them of this sickening crime Five months later in this crazy that there's weirdo suck out here in the world do stuff like this Like you can't even be a beautiful young lady in 2000 like in this world at all Without having to worry about stuff like this like it's crazy. You know, you can't you can't even defend yourself either It's like in Tunbridge Wells another horrifying crime occurred at around midnight on the 24th of November 1987 a 20 year old woman named Caroline Pierce had just taken a taxi back to her home She had been on a night out with her friends and was making her way inside as the taxi pulled away Neighbours heard screams around the time she got home. They looked outside, but they couldn't see anything The next day Caroline never arrived at work When her family and friends were unable to contact and locate her panic ensued Memories of the grim case of Wendy entered the public consciousness With fears that a serial killer was now at large The taxi driver was questioned, but it was clear that he played no part He also said that he didn't see anybody suspiciously lurking around the area when he dropped her off He claimed that he didn't even see anyone at all Three weeks after Caroline had gone missing on the 15th of December a farmer found her body Decomposed in a drenched itch and it seemed like she had been there for some time These murders would probably been there tonight at the same night three weeks They found it decomposed already Um known as the bed-sit murders and they were unsolved for decades But in 1999 advances in forensic science allowed the investigators to get a full DNA profile of Wendy's killer DNA that for the first time linked both murders forensically Oh, yeah, because you know Because he he forcibly Forced their himself upon them, right? Police found 1000 names on a national DNA database Which could have belonged to relatives of the samples taken from the crime scene and whittled this list down to 90 priority names After voluntary samples had been taken from the list of 90 a relative of one of those people was found to have a near identical match Having evaded justice for 33 years. The man responsible for these crimes was arrested on the 3rd of December 2020 So three years ago Two years They didn't even get a man a chance to reply they ain't asking to come in or nothing They was in uh, we know it's you buddy He know what's up. He know what he did talking about Oh, it ain't even bright in there. It ain't no more brighter than what you had it in there This is what i'm gonna say. Yes, and we're investigating the murders of Wendy Nell and Caroline Pierce in 1987 As part of that investigation you've been linked as a suspect both geographically and criminally All right, David you're under arrest on suspicion of the murders of Wendy Nell and Caroline Pierce in 1987 John Brode didn't even blink He just stood there like dang they got me, you understand You do not have to say anything but it may harm me to fence if you do not mention when questioned something which are later lying in court. Anything you do say may be given an evidence. You are being arrested. So secure and preserve evidence by means of questioning. So we can conduct searches, so forensic samples can be attained and to prevent your disappearance.  David Fuller. David was a 66 year old father of two who worked as an electrician at a hospital. Photographs that were taken in the 1980s were found in David's house. They showed him wearing the exact style of trainers that would have left behind the shoe print on Wendy's blouse. However, the full extent of the horrors of what this man did would soon come to light. Officers had no idea how to print. I feel like everybody probably had those shoes back in those time. But you know it's all the stuff combined that got him in there. If this individual actually was, David was questioned and his house was searched. There's some stuff stuck on the back of that one as well. I've got the body worn on it. It literally has it's happened. Well, on the back of his chest to drawers, it looks like he's got some hard drives in there while it's like put this stuff to the back of his chest to drawers. So who's prepared for this day? Just in case, let me do this, this and the third to try to hide it. It's the sand disk one that I can read there, with the fronts there. He admitted to the murders of the two women, but they also discovered that the pin number to his bank account was 1987, the year he had killed them both. After David's arrest, the police uncovered four hard drives with five terabytes of data storage in total attached the back of a cupboard. These hard drives were well hidden because on them were four million CP images, mostly downloaded from the internet. And you know what you're thinking? And no, it's not a mistake, four million. They also found 818,000 images and 504 videos of David documenting himself having intercourse with 100 corpses. Some of the names are... Yo, what? Wow, he's one of them. These folders were labeled Necrolord and the best yet... Necrolord? Yet. He had even kept a secret diary where he documented his twisted crimes. He had been working for the NHS and he had used his swipe card to gain access to the hospital mortgaries. He would wait until the hospital staff had gone home. He would then creep inside these rooms, have intercourse with the bodies and film himself. The victims included a nine year old girl, two 16 year olds and a 100 year old woman. In total there were over 100 victims. After he was done he... 100 victims? He then used the mortuary records to lock their details into his own notebooks and then look up his victims on Facebook. David was arrested and charged with murder and violating corpses. How many years he get? This is top weirdo of the century. I've never seen or heard of something weirder in my life. It was always in the evenings, David. It was always evening time. Can you tell me what you've been doing? What's up, David? Can you try and explain it to me? I. You find it hard? Yes. Thank you. Well, can you tell? No. Okay. I guess. We know that. So admit it. I am admitting the offenses. But I don't really want to go into detail. Yeah. What's offenses you admit, David? Yeah, at the end of the day he did it. He's saying he did it. I don't even think we really want to know the details. That's nasty. Courses, dead bodies? Analyzed bodies? Just describe to me. My bad, man. Hopefully I'm not over mumbly sounding. Because I got this mask on, man. I got a cut on my face. Pretty big cut. I find it disgusting. So therefore, you know what I'm saying? I ain't going to put it on the internet. In terms of these two penetration corpses. Okay. Do you know how many occasions, nobody? David, lost count. David, I was asking where you put the hard drives. Where you would have kept them. Can you help me with where you kept them? You said it was in the office. Now you've got a desk in the office. Yeah. And you've got a wall drive. You keep quite a lot of your clothes in the office, haven't you? Yeah.  Now we've found some hard drives. There was a package to the back of a small chest of drawers. Does it help you? David? Yeah? Is that where you keep the hard drive? Yes. Okay. The second part. I feel like in this interrogation, he's like reliving every time he did it. That's this, David. It's the recording, isn't it? What's been happening? Okay. We'll have to go through that in a little bit more detail. Just for now, David. All right? Just for now. All right. When that's been happening, okay? All right. Don't be embarrassed now, David. Don't be camera shy now. Have you been recording yourself doing those things? Like, so tell me in the comments, man. What's your opinion, man? Is this like a mental thing? Or was he down bad so horrendously? He never got no play that this is what he had. He got two kids, don't he? Look, recording yourself. He penetrates in the corpses. Yeah. What did you retain the recordings for? Well, scratching his head. He can't believe himself. You can't believe yourself. We can't believe what I'm hearing either. NHS. Once again, somebody you're supposed to be trusting. He in there downstairs in the morgue after hours. Going crazy. That's sad, man. What about the families of these victims, man? I hope ghosts are real. They come back and hunt that man. Was it for virtual pleasure, David? In the same way as somebody would keep. Okay. So it started then. And it's a couple of times. How many would you say? Not many. Not many. And in terms of at the Kensington Sussex Hospital, were there any occasions other than at those hospitals? So it's always just been a hospital setting. It's not been at any funeral parlours or anybody that you knew that had died or anything like that. Okay. Do you know why you started? Yeah, that's a good question. Do you know how you started? Like, what clicked in your mind where you was like, you know, I got to. I got to have me some corpse. Some corpse box is crazy. Do you remember the first time? You remember the first time? David, do you remember the first time? Get closer so y'all can hear me. Did you record who they were? Any details? So are we classifying them as a pedo too, right? Because these bodies are nine years old and... Did you record their names? Yes. Did you record their ages? Yes. They got the whole, you did? Oh, they sent them to me and everything? Okay, I hope you get life, like natural life. David, this is probably the hardest question. You know what? Well, the hardest one for you to answer. What were you trying? I feel comfortable answering it. Why did you record what was going on? Loki, he probably just sent it on the dark web or something. Or to play it back for his own gratification later. Like, imagine, Bro went in there and had a tripod, set up a tripod, camera. You know what I'm saying? Probably lit some candles, went crazy. Oh, that's disgusting.  That's nasty. I can help you because you know we've searched your office. You can't help, huh? We've seized your office and we've downloaded computers. I don't know, David, whether there's some things in your life that have given you the opportunity to do it. Can you see where I'm coming from? That could be a particular shift. Or it could be a particular day when the mortuary closes early. It could be something else in your personal life that allows you to do that. Or there may not be a sequence. It might just be when one of the females enters the mortuary. What was your thought process about your fending, David? How would you decide that a particular offense would occur?  I didn't. Upon learning of this horrific news, David's wife... Bro, no. Oh, he had a wife, too? He knew exactly. He knew to answer all of them questions. They're just too embarrassing to answer. Upon learning of this horrific news, David's wife left him. She was, of course, also distraught at what he had done. That's gotta be hard on her. Imagine being somebody's wife who would regularly take pleasure in doing dead bodies before you. No offense, but it's nothing that she could have done about it. It's something that's going on in his head. Imagine what's going through her mind. That he didn't also distraught at what he had done. Those who knew David stated that he didn't seem like the kind of person to do such a thing. They described him as a normal and quiet person. David initially denied the murders on the grounds of diminished responsibility before unexpectedly changing his plea to guilty on the fourth day of his trial. He pleaded guilty to murdering Caroline and Wendy, as well as 51 other offenses, including 44 charges relating to 78 victims and the mortars between 2008 and- Oh, so yeah, repeat victims. November of 2020, David was sentenced to life in prison without the possibility of parole. Well, time! That's the first time I've ever heard that out of the UK. Life without the possibility of parole. That's W. W conviction. The systematic nature of the way you had recorded your offending can be illustrated by reference to one portable hard drive, Exhibit KH83. That contained three folders, Necrolord, Register and Deadly. Some of the content could not be accessed. Within the subfolder, Deadliest, this also contained subfolders. One of the subfolders, titled 00 Best Yet, contained a further 36 folders. Of these, 27 were titled using a number, then a woman's name, and then a further number.  It is by looking at such images, comparing the creation dates and other records with the mortuary logbook. Like the judge and everybody involved not discussing, like, how do you go on? Like, how do you get past stuff like this as a judge? There are books and other evidence that the identities of the 91 named victims were found. It would be wrong to bring any victim, and it is unnecessary to do so. Among those you abused were women who had led fulfilling and rich lives. They were the best. One had flown across the Atlantic in a propeller plane via Iceland. Another was a talented skier. One had worked at Bletchley Park during World War II. Many had long... One worked at Bletchley Park down World War II. Oh, yeah, they did say there were some 90-year-olds in there. ...long and happy marriages. A number worked hard in professions such as... ...or in the NHS, caring for others and looking after the interests of their students, clients and patients. All had families that they nurtured and loved. Some spent the last part of their lives in suffering, whether through disease or old age. But they didn't lose their dignity until you decided to take it from them. At the time of the sentence... Insane. ...saying 81 of the mortuary victims had been identified. However, due to difficulties in identifying all of the corpses he violated and fearing they may have been hundreds of others going back to when David's hospital employment record began, Kent Police set up a call centre at the conclusion of the trial to deal with any public concerns. The government also announced a public inquiry while the NHS England ordered all hospital staff to review their security. In December of 2022, he pleaded guilty to essaying the bodies of 23 more women, bringing the total of mortuary victims to 101. For this, four further years were added on to his sentence. As of right now, no evidence has been found of any further victims. DS David Shipley of the Kent Police investigation team was tasked with analysing the pictures and videos. He said, David Fuller committed every intimate act that is possible to do with the human body. What a guy, he was doing everything, foreplaying everything. There was a complete lack of emotion being shown by him. It seems to me that there was no thought whatsoever for the fact that it is someone's loved one.  There wasn't an ounce of humanity demonstrated by him at any point. It took David Shipley over one year to watch all of the videos that David had recorded and I can't even imagine how disturbing this job must have been. That's what I'm saying, how do you go back to your everyday life after watching stuff like this? He also said that whilst reviewing these videos, he saw that David was almost caught naked and in the act. Someone had walked in but they didn't see him. After this person left, David didn't stop. He continued to violate the victim's body. He was even violating the bodies of these women and girls up to two to three days before he was arrested. It's likely that he only stopped murdering because he had access to the mortars in the hospitals. After learning what happened to their beloved relatives, the families were deeply disgusted and outraged. At least one of the relatives even took their own life after learning of the despicable crimes of David. Tanya Macalden died in 2017 following a battle with breast cancer which of course would have been awful enough for the family. But years later, after suffering from the loss of Tanya, they were told the horrifying news that her body was violated by David. Tanya's son, 24 year old Tom, has said that learning this news left him feeling anger, sadness, sickness all rolled into one. He said whenever he saw a picture of his mother, an image of David would pop up in his mind. In Easter of 2014, a family from Paris went on a trip to the United Kingdom. Shortly after arriving, they were involved in a collision. Mary, who was just 16, and her sister Helen, who was 22, died instantly. Do you suffer from anxiety, feeling irritable or moody?  Like that's nasty, bro. Their father, Michael, who was also in the car, passed away two weeks later. The mother, Nikki, was the only survivor. Years later, she learned that her two daughters had also been violated by David. She describes what he did as demonic and satanic, but says that she would also forgive him. She has said that she would like to speak to him, to tell him that he needs Christ. A woman named Jane Knight died from a stroke at the age of 100. She is believed to be David's oldest victim. Her son came forward and spoke with the media after the NHS trust was offering 7,500 in damages with possible additional payments up to 15,000. That's not enough. Recognition of the psychological damages caused to the victims' families. Jane's son said, I've been having trouble sleeping. I've been struggling to get the thought of what he did to my dear mum out of my head. I don't think any of us will get closure till we get the justice we deserve. I wonder how he's doing in the prison yards, man. What's going on in there? He said he was aghast when he was told of the 7,500 pound sum. He went on to say, I didn't want to come across as being a mercenary, but I believe my mum and the families need justice, and they also need compensation. Towards the end of 2022, an agreement was reached. A lawyer representing the families of the victims said, The trust has agreed to put in place a voluntary compensation scheme, but they do not admit that they are legally liable to pay damages to the close family members of the deceased victims of David Fuller. They want to protect themselves. They want to protect their image. Solicitors believe that the average claim would be around 25,000 to 30,000, although a small number could get six bigger sums. The case of David Fuller is believed to be one of the worst in British legal history. The first video I made on David had a number of commenters saying that the victims were already deceased, so it didn't matter as much.


Bill Smarzo Studio B wanted to ask a few questions. Walk us through a day in the life, Bill Smarzo.  So I like to tell people I have the best job at EMCU. I get to spend all of my time talking to customers, right? Working with them and figuring out where and how to start their big data journey. So it's a, my average day is if I'm not actually traveling someplace in exotic locations like Des Moines, Iowa, and Omaha, Nebraska and Chicago and places like that. So if I'm not actually traveling and working with customers, I'm actually on the phone many times actually reviewing stuff, reviewing processes, trying to help them figure out where are the right business cases to go after?  And you know, you've talked about looking for business-led, business opportunity-led IT deployments. Tell us a little bit more about that. So the challenge that we've seen for the last, I would say the last 18 months is that for a lot of organizations, big data has been nothing but except technology. It's almost been a science experiment. And what's happened over the last four or five months is the business has gotten clued on to the fact that I have this wealth of data, both structured and unstructured formats. I have the ability to ingest this data in real time, and I have some massive analytic capabilities available. I can start teasing insights out of that data, insights about my customers, my products, my operations that really can help me drive better decisions and even create all new monetization opportunities. What has been some of the aha moments that your customers have when they finally grasp that and really look forward to the journey of themselves? So I think the area that surprises them most is that they're very heavily focused on trying to figure out how do I optimize my existing business processes, which by the way is a great place to start for a lot of organizations. But the aha comes when they start mining all that insights that come out of that process and realizing, oh my gosh, I have all these great insights that I can monetize in new ways through new products, new services, new user experiences, even package and insights and sell it to others. So that aha moment comes after they start of doing this optimization process and realizing that the byproduct is all these insights that's really valued by other parties. How can you kind of try to fast forward to that point? How can you, as you're educating people and businesses, how can you get them to really get to that moment sooner than they are in some cases? Well, it takes a corporate mindset on their part that really is about joining together and driving collaboration between the business side and the IT side. The groups that I've seen successful are the ones that bring both parties to bear and then we run a series of envisioning exercises with them to help sort of to imagine the realm of what's possible. Because they haven't really thought through, you know, I have all this unstructured data, consumer comments and notes from call centers and email dialogues and in those unstructured data sources are invaluable nuggets that can help them learn more about their customer's interests, passions, affiliations and associations that can impact the way they interact with customers. So the way that we do it is we can really drive collaboration between the business and IT. It's to run these envisioning exercises to really help people to sort of to envision the realm of what's possible. Are there any unique kind of case studies or just big ideas around these business opportunities that where it makes sense for big data that you maybe hasn't been implemented but has just been, you know, a really big idea in that realm. Well, I would say we're seeing case studies kind of popping up in two different areas. Number one, companies that are dealing with consumers. So retailers, banks, teleco companies, they're the ones who can get a lot of value out of their subscriber and customer behavioral analytics, right, understanding what kind of things their customers do, what kind of products take a light, how do they behave in certain situations? So that's a huge wealth of information, not only from a customer engagement perspective, but also from a developing and marketing products perspective. On the B2B side, the business to business side, we're seeing a lot of interest in predictive maintenance. So doing a project with an organization, they have wind turbines across the great state of Iowa and they're trying to figure out how do I optimize performance of those turbines and how do I do better predictive analytics? So again, business to business organizations are trying to figure out, I have all this censored telemetry data, how do I mine that to really help improve my maintenance predictability and drive better performance of all my operations? Fantastic. Mr. Smudger, thanks for stopping by at B2B. Yeah, thanks.


Welcome back to our lecture series Math 31-20, Transition to Advanced Mathematics for students at Southern Utah University. As usual, be a professor today, Dr. Andrew Missildine. In lecture two, we are going to talk about the very important topic of sets. Advanced mathematics is essentially, its foundation is what we call set theory. So it's very important in the second lecture that we establish what does mean when a mathematician talks about a set. Well, our definition is that a set is a well-defined collection of distinct objects. So proverbially, you can think of a set as a bucket that holds stuff. It's a collection of things, and those things inside the set, we do consider them to be distinct objects. We'll get a little bit more into this in a little bit here, but we don't allow repetition to set. If the thing's in there once, it's only in there once. All the objects contained in the set are considered different. In fact, the objects that belong to the set are typically referred to as the elements of the set. Now, we'll get into this at the end of lecture two a lot more, but I do want to say something about, what does it mean to be a well-defined collection? It turns out that not all collections are sets. One thing we will talk about in a future lecture, in contrast to a set, there's something called a list or someone might call it a string, where the order of the elements matters, like there's a first element, second element, third element. In a set, the elements are unordered. Is there repetition allowed in your object, like maybe in a sequence, maybe the things could happen more than once? So a set is not just any type of collection of things, but what does this word well-defined here mean? Like I said, we'll get into this more at the end of this lecture when we talk about Russell's paradox, but for the moment, we can say that well-defined means that there's a rule that enables one to determine whether a given element belongs to the set or not. So there has to be a way to decide, does object A belong to the set or does it not belong to the set? And so we have to know, is this element, is this object an element of the set or is it not an element of the set? That decision has to be possible and a set will be well-defined so that there is always some rule or property, algorithm, whatever, that determines whether the object belongs to the set or not. Now it is possible that you can have an empty bucket, that is, there do exist sets that have no elements whatsoever. And honestly, there's only one set that contains no elements and this is referred to as the empty set for obvious reasons. And this might seem weird that we would care about an empty bucket, but nonetheless, the empty set shows up with such high frequency that it does deserve some attention. And we're gonna introduce notation for the empty set. Typically when we draw the empty set, we're gonna draw what looks like an O, maybe a zero, but with a slash through it. You might be used to this in the past, like if you were solving an equation, like in an algebra class or maybe calculus and there was no solution, you would often write this symbol to represent no solution. What actually you were writing that circle or oval with a slash through it, what you were saying is that the solution set, that is the set of solution, the collection of solutions to the problem, that solution set is empty and hence we use the empty set to commonly denote things like no solutions to a problem. Another notation you could use, this one's far less used, but if you just do two curly braces with nothing in between them, that also denotes the empty set. This notation will make a lot more sense in a little bit because as we talk about sets, we often denote sets using curly braces and the things between the curly braces are then the elements of the set. Well, if there's nothing between the curly braces, it's an empty set, but in particular, this symbol right here is the more commonly used symbol. Let me give you some examples of sets. So one set that we might talk about in a class like math 3120 would be the set of all even numbers. This set would include things like zero, two, four, six, eight, who do we appreciate? But it would also include negative numbers like negative two, negative four, negative six, et cetera. All right, and so I'm already kind of suggesting how sets look like. We often start and end a set with these delimiters, these curly braces to indicate that the things between the braces are in fact the sets. And then we can list the elements of the set, but in this case, the set of even numbers, there's an infinite number of elements. I can't possibly list them all. So sometimes some dot, dot, dots, these ellipses will be used to indicate that there's a pattern established and then continue on in that pattern. So these are the types of sets we talk about all the time in mathematics. But to be fair, to be a set, the elements of the set themselves do not have to be numbers. We could talk about the set of all books about travel to Chile. I've never been to Chile and asked such. I've never traveled there. I haven't read a lot of books on there. So I actually don't know very many books about travel to Chile. But I mean, that is a set. We could look it up online or go to a library and ask all the books they have about traveling to Chile or such. Clearly this has to be a finite set, which of course in this situation that means that the number of elements of the set is a finite number. The nation of Chile has only existed for a finite amount of time. And as people have written about it, it would be impossible for a person to write an infinite number of books in a finite lifetime. So whether this could, I mean, this could be, there's probably thousands, maybe even tens of thousands of books, I don't know, but it certainly had a finite collection here. But that is in fact a set of objects. The objects of the set are books, books about the specific topic here. And so we can decide whether a book belongs to this set or not. If we have a book that's like top 10 tourist traps of Chile, well, yeah, that's a book that belongs here. But if we have another book, like say your textbook to this class, the book of proof, for example, it's like, no, that would not belong to the set. That's not about travel to Chile, unless of course you're considering this sentence. No, that's not the case. We have to be able to decide whether it belongs or not. And our textbook does not belong to the set, all right? Now this next one's meant to be, of course, a fun example here. Consider the set of all purple unicorns named Hank, whose image is currently contained on the screen. And look around for Hank here. Where are you, Hank? I don't see Hank anywhere. Oh, there's a unicorn. But wait a second, this isn't a purple unicorn. That's an orange unicorn. And even if this unicorn was purple, this unicorn's name is Buster. It's not Hank. And so when you look at this set described in part C right here, the set of all purple unicorns named Hank, whose picture is on the screen, there are no purple unicorns named Hank depicted on the screen right now. So this third set is actually the empty set. And so this actually connects to the idea I was mentioning earlier when we talked about the empty set, that the empty set shows up all the time in the study of set theory. Now, when we describe this set C, we didn't say this is the empty set. We actually gave a property that determines whether something belongs to the set or not. And because the property was so restrictive, purple unicorns named Hank on the screen, it turns out that no object satisfies the property of the set and therefore the set was empty. We got things that were close to it. Buster really wants to be part of the set. But alas, he doesn't satisfy the property and does not belong. So oftentimes when we describe a set, it actually might be an empty set based upon the property. And so therefore oftentimes when we work with a set, we actually have to prove that it's not empty. The property itself might be vacuous. That is, there might be nothing that actually satisfies the condition like our purple unicorn named Hank here. Like I mentioned on the previous screen, sets are often denoted using these curly braces. Here's a computer type set version of those curly braces. They look a lot better than my handwriting here. There's two common ways that one describes a set. And one way is just to describe all the elements in the set. So you start off with one of these braces and then you list all of the elements in the set. So like X1, X2, X3, X4 up to XN for some number here. N is meant to be an arbitrary number. So it could be 10, it could be 1 billion, it could be 5,280, what have you. And so because I don't necessarily wanna list all of them, particularly because I actually don't even know how big N is in this situation. Again, we use these ellipses to denote there's a pattern established here and continue the pattern. Now, one should always be cautious with ellipses right here because it depends on a pattern. When I use ellipses like this, I'm taking faith in the audience that the audience will understand the pattern that's established here. If your pattern's insufficient, you might not be able to get away with that. Like if I'm like, oh, here's a sequence of numbers that begins one, two, dot, dot, dot. It's probably not fair to assume the audience knows what the next numbers are. I mean, are these the positive integers that we're trying to establish here? That starts with one, two. We could be like one, two, four, eight, 16. It could be like a geometric sequence where we're increasing by powers of two. But we could be like one, two, six, 24. It could be like factorials. One factorial is one, two factorials, two, three factorials, six. But maybe it's some type of like Fibonacci sequence or something like one plus two is three, two plus three is five, three plus five is eight. So if you're gonna use an ellipse to try to describe a pattern or trend, it needs to have sufficient context and sometimes sufficiently many elements so that it's clear to the audience here. Now, in this example where the X1s, X2, XNs, they're not specified, all this means to the audience is that there's N objects contained between this. Now, if we wanna describe a set, we can't actually describe every element that belongs to the set. And there are some advantages to this that determining whether someone belongs to the set or not is super easy, because it's like, oh, does X2 belong to this? There it is, it belongs. Does this object belong to the set? Nope, it's not in the list, so it's not there. Now, that works if you have a small set with particularly finite and maybe not a lot of elements. If you get lots of elements, if you have like a billion elements that takes time to search the billion objects to make sure whether something belongs or not. So actually in practice, more often than listing the elements of a set, it's more common to use what's called set builder notation, which set builder notation will look something like this, for which our set X is defined, again, you start and end with the curly braces to determine that there's a set in play here. Then the first thing you see will be a typical element of the set, some generic arbitrary element of the set. And then there'll be some type of division here where a lot of people use a vertical line, some people use a colon, both are commonly used, there's not one preferred over the other, I particularly use a line to separate this. And then after the line, so this is our typical element, after the line, then we say that the things that belong to the set will be those things that satisfy some property. In this case, we're gonna call it generically property P. So the elements of my set capital X, or lowercase X, such that X satisfies some property. So there's some property that determines whether you belong to the set or not. And so then you look to the property to then see whether X belongs to the set. If X has property P, then it belongs to the set. If X doesn't have property P, then it doesn't belong to the set. Now, like I mentioned earlier, there are some things that you need to take into consideration when you compare sets to other types of collections. Inside of a set, repetition is irrelevant. If a set is inside, sorry, if an element is inside the set once or twice or three times, we make no distinction between that. Repetition is not considered with regard to a set. So consider the following two sets. The set that contains the elements one, three, two, two, and the elements, the set that contains the elements one, three, two. These two sets are considered to be equal as sets. The fact that two shows up twice in the first set makes no bit of a difference compared to the second set because all that matters is that this set contains the element one, the element three, and the element two. If there's two reasons why it contains two, it doesn't make it any less true that two is inside the set. When you look at the second set, it contains one, it contains three, it contains two. And because the sets contain the same elements, they're the equal as sets. The repetition is irrelevant for a set. Another thing that's important when you consider sets is that the order in which the elements occur does not matter. The order or arrangements of the elements is irrelevant. So continue with this example here. If you take the set that contains the elements one, two, three and you compare that to the set that contains the elements one, three, two, those are considered the same sets. It doesn't matter that as I'm listing the sets, three shows up before two in the second's representation, they're both the same set. And likewise, the set that contains three, two and one, these are, this is equal to the previous two sets. These are all the same thing. All that matters when it comes to a set is the membership. Who belongs to the set? When you look at the first set, it contains the elements one, two and three. When you look at the second set, you have the elements one, two and three. When you look at the third set, you have one, two and three. While they're listed in different order, the membership is the same. All three sets contain the numbers one, two, three. None of the sets contain four, five, six, et cetera. They're the same set. Order doesn't matter, repetition doesn't matter when it comes to a set. All that matters when it comes to sets is the membership. And so since membership is king, when it comes to considering sets, let's actually introduce notation to talk about set membership. When considering if an object is an element of a set or not, we use the symbol right here. Kind of looks like an E. It looks more like the Greek letter epsilon here, but this here is understood to mean in, okay? That this symbol means that an element is in the set. So when you see this symbol, it means such and such is an element of the set. And likewise, if you see that symbol with a slash through it, that means the object is not an element of the set. It's very common in mathematics that when you have some type of relation symbol, like in this case, A is inside of X. If you put a slash through the relation, that actually negates it. So putting a slash through something typically makes it not. Like if you say something like, oh, one plus three is equal to four, you have this symbol, this relation equals. Well, on the other hand, one plus three is not equal to five. You put a slash through it, it makes it not equal. Same thing applies in this situation as well. So consider the set A, which contains the four integers one, two, three and four. It is true that two belongs to the set A. So we would say two is in A, two is a member of A. Two belongs to the set. It's likewise true that one belongs to the set and four belongs to the set. But we can also say that five does not belong to A. Neither does the number pi, it doesn't belong there. The square root of two is not in there. We also have that like Pikachu, cute little Pokemon there. Pikachu does not belong to the set. The set only contains one, two, three, four. Anything not listed here does not belong to the set. And so we could say something like this. Now, of course, one begs the question, why would you have ever considered whether Pikachu belongs to that? Isn't it a set of numbers? This is leading to the idea of a universal set, which we'll talk about more about that in the next video for this lecture. But as we end this video introducing this to sets, I do wanna introduce us to five very important sets, some of which we've already talked about in this lecture series already. These are symbols that are commonly used within mathematics to denote these sets right here. So this font right here, one thing that I should mention in mathematics, we use different fonts to mean different things. So if I have like a lowercase X versus a capital X, they probably mean different things. And if I start changing the font of my X, I might mean something else as well. So you should always be careful about the font. They oftentimes will mean specific things. The font you see on the screen right now is commonly referred to as the blackboard font. I suppose the idea was that when you write these symbols on a blackboard in a traditional lecture, this is the font they use to denote these things. The very first one, blackboard N, we'll just call it N, of course, notes the set of natural numbers. So N belongs to the set if N is a natural number. So the natural numbers, of course, include the pods of integers, one, two, three, four, et cetera. And in our lecture series, remember we've talked about this before, we include the number zero as a natural number. Some people might exclude zero, but we always include zero in our consideration natural numbers for this lecture series. This symbol right here, this is blackboard Z, or we'll just call it Z for short. If you wanna go European, you can call it Z, if you want. But Z here denotes the set of integers. So N belongs to the set if it's an integer. And integers include all of the natural numbers, including all these negative, they're additive inverses as well. So you get the negative integers, pods of integers in zero, like so. To give you a little bit of explanation of these symbols, clearly blackboard N makes sense, and it's for natural numbers. Where does the Z come from? That actually comes from the German here. The German word for number, I'll probably mispronounce it, here's all. That's where the symbol Z comes from in that situation. Q makes sense. Q denotes the set of rational numbers. A number belongs to this set if it's a rational number. Remember rational number is a quotient of integers. That's where the Q comes from, Q for quotient. You get a quotient of integers, P over Q. So P and Q are both integers, but you also have to require the denominator cannot be zero. So fractions like two over zero are not rational numbers. All right. Of course, the word rational, where does it come from? The idea is it's a ratio. Ratio and quotient are synonyms. And so the rational numbers are those numbers which can be written as a ratio or a quotient of integers. Next, I wanna mention blackboard R, or just R, is gonna be the set of real numbers as defined like in a calculus class. So this will include rational and irrational numbers there. And then blackboard C, this is commonly used to denote the set of complex numbers. Remember complex number is a number of the form A plus BI where A and B are potentially any real number and I is the algebraic square root of negative one. As we've probably seen before in like calculus or pre-calculus or things like that. These five sets will be very important to be aware of. And these are very common symbols. Whenever you see these symbols, rarely, rarely will they be anything other than the five things you see on the screen. Now, one other comment I wanna make about writing these things is that when you use like a computer, you would type this up in latex. There are specific symbols here you would use. You would use the backslash math BB font. BB stands for blackboard, the blackboard font here. But as one often writes these, it's sometimes difficult to write these symbols for the natural numbers. Typically what someone does is they'll write just a capital N and they'll add an extra line here to indicate the blackboard font. And that's typically how you do these things like for the rational numbers, you draw a Q with an extra line, the real numbers with an extra line for the R there, complex numbers of C with an extra line. These are very common how they're written. For the Z though, there's a little bit of controversy there. Some people when they write the Z, they just kind of follow the pattern of the other one. They write a Z with an extra line in it. This is acceptable.  That actually looks better, more accurate. And so some people take issue with that. This is generally how I write the Z, but I'm not picky about it, but that's how I've learned to write it mostly because some people were picky about it. I'm not sure why Z is so exceptional in that regard because we look at like N, it doesn't match the font. Like the blackboard is actually in the diagonal there. So it'd be something more like this. Whoops. That's actually the more correct symbol, but I don't actually know many people who are picky about that. When it comes to the rational numbers, it's more like you have, oh boy, I'm not sure I could even do this on the fly. You get something like the following. That's more accurate. So don't be too worried about these. These are all acceptable fonts you could use if you have to write these things by hand.