 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. But some people are very insistent that when you write the Z, you actually have to draw like a seven and then an upside down seven. 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.